newPackage( "MatFacCurvesP4", Version => "1.0", Date => "October 25, 2016", Authors => {{Name => "Frank-Olaf Schreyer", Email => "schreyer@math.uni-sb.de", HomePage => "http://www.math.uni-sb.de/ag/schreyer/"}, {Name => "Fabio Tanturri", Email => "fabio.tanturri@univ-amu.fr", HomePage => "https://www.i2m.univ-amu.fr/spip.php?page=pageperso&nom=tanturri&prenom=fabio&lang=fr" }}, Headline => "Matrix factorizations and curves in P^4", PackageImports => {"BoijSoederberg", "kGonalNodalCurves" -- available at https://www.math.uni-sb.de/ag/schreyer/images/data/computeralgebra/kGonalNodalCurves.m2 }, DebuggingMode => true ) export { "matrixFactorizationFromModule", "idealFromMatFac", "isSmoothCurve", "h1NormalBundle", "alexanderSurface", "linSysAlexander", "properTransformAlexander", "auxiliarLineBundle", "singularCurveInP4", "curveGenus12Degree14InP4", "curveGenus13Degree15InP4", "randomCurveGenus10Degree13InP4", "randomCurveGenus12Degree14InP4", "randomGenus12Degree8CoverOfP1", "curveOnAlexanderSurface", "isInBoijSoederbergCone", {* "bettiCandidates", "bettiCandidatesIrrelevant",*} "listOfPossibleMatFac", "candidateTables", "candidateTablesWithIrrelevantIdealAssociated", "arisingMatFac", "precompiledListOfCandidates", "verifyAssertionsOfThePaper" } ------------------------------------------------------------------ -- The main functions of the paper -- ------------------------------------------------------------------ matrixFactorizationFromModule=method() -- Input: a module or an ideal -- Output: a matrix factorization induced by the module on a hypersurface -- supporting the module or contained in the ideal matrixFactorizationFromModule(Module) := N -> ( S := ring N; gIE := gens annihilator N; X := ideal(gIE * random(source gIE,S^{-min(degrees source gIE)})); SX := S/X; fNX:= res (N**SX); c := # flatten entries vars S; return (fNX.dd_c**SX^{first min degrees target(fNX.dd_c)}, fNX.dd_(c+1)**SX^{first min degrees target(fNX.dd_c)}) ) -- matrixFactorizationFromModule(Ideal) := I -> ( S := ring I; return matrixFactorizationFromModule((S^1)/I) ) TEST /// S = ZZ/32009[x_0..x_3]; I = minors(3,random(S^4,S^{1:-1,2:-2})); betti res I (phi, psi)=matrixFactorizationFromModule I; betti res ((S^1/I) ** (ring phi)) betti phi betti psi SX = ring psi; phi*psi /// idealFromMatFac=method(TypicalValue=>Ideal) -- Input: a matrix (one of the two maps of a matrix factorization) -- and a BettiTally encoding the LHS and the RHS of the monad whose -- central element is /mathcal{F}, in the paper -- Output: the ideal of the curve obtained as homology of the monad idealFromMatFac(Matrix, BettiTally):=(psi,monad)->( SX:=ring psi; psi=psi**SX^{first min degrees target psi}; phi:=syz psi; bSX:=SX.baseRings; S:=last bSX; X:=ideal SX; targetMonad:=apply(select(keys monad,L->first L == 0),L->last L); degTargetPsi:=degrees target psi; degMonad:=apply(select(apply(#degTargetPsi,i->{i,degTargetPsi_i}), C->any(targetMonad,l->l==last last C)),C->first C); A1:=psi^degMonad; A2:=syz(A1,DegreeLimit=>first max degrees source psi); sourceMonad:=flatten apply(select(keys monad,L->first L == 1), L->toList(monad_L:last L)); A3:=A2*random(source A2, SX^-sourceMonad); A4:=A3|phi; A5:=syz transpose A4; beta:=syz transpose (A5_(toList(0..rank source A5-2))); alpha:=transpose(A5_{rank source A5-1}); IXC:=ideal mingens ideal(alpha*beta); return sub(IXC,S)+X ) --- TEST /// p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; time singC=singularCurveInP4(S,12,14); omegaSingC=Ext^2(singC,S^{ -5}); -- canonical module of C fomegaSing=res omegaSingC; sM=S^{ -5}**coker transpose fomegaSing.dd_3; (psi,phi)=matrixFactorizationFromModule(sM); betti psi monadShape=betti map(S^{2:-1},S^{2:-1,2:-2},0) IC=idealFromMatFac(psi, monadShape); betti res IC /// isSmoothCurve=method(TypicalValue=>Boolean) -- Input: the ideal of a curve -- Output: true if smooth, else false isSmoothCurve(Ideal) := C -> ( S:= ring C; if not (dim C == 2) then error "expected the ideal of a curve"; -- if embedded points then n:=dim S; if n<=4 then ( -- check for no associated points if not dim Ext^(codim C)(C,S) <=0 then (use S;<<"there are associated points" <4 then ( projectionCenter:=ideal(apply(4,i->S_i)); if not dim(projectionCenter+C) ==0 then (use S; <<"bad projection center" <S_(i+4)))); if not degree C1 == degree C then (use S; <<"bad projection center" < ( con:=prune(IC/((IC^2)+X)); -- conormal bundle S:=ring IC; SC:=S/IC; N:=Hom(con**SC,SC); return HH^1(sheaf N) ) TEST /// S = ZZ/32009[x_0..x_3]; I = monomialCurveIdeal(S,{1,2,3}); X = ideal(gens I * random(source gens I, S^{1:-min(degrees source gens I)})); time h1NormalBundle(I,X) /// alexanderSurface=method(); -- Input: S, the coordinate ring of P4 -- Output: (Y, pts), where Y is an Alexander surface in P4 and -- pts is a list of the ideals of 10 points used to blow P2 up alexanderSurface(Ring) := (S) -> ( y:=symbol y; R:=coefficientRing(S)[y_0,y_1,y_2]; -- coordinate ring of P2 -- We construct Y as the blow-up of P2 along 10 general points, -- embedded in P4 via the linear system |13L-4(p_1+...+p_10)| pts:=apply(10,i->ideal random(R^1,R^{2:-1})); Hs:=((gens truncate(13,intersect(apply(10,i->(pts_i)^4))))_{0..4}); bUp:=map(R,S,Hs); return (sub(prune kernel bUp,S), pts) ) TEST /// S = ZZ/32009[x_0..x_4]; time (Y, pts) = alexanderSurface S; betti res Y /// linSysAlexander=method(); -- Input: a pair of integers (g,d) -- Output: for every admissible pair, the corresponding linear -- system L as in Table 5.2 linSysAlexander(ZZ,ZZ) := (g,d) -> ( cases:={{16,17},{17,18},{18,19},{19,20},{20,20}}; if all(cases,c->c!={g,d}) then error "case not implemented"; L:=0; if g==16 then L={21,7,7,7,7,6,6,6,6,6,6}; if g==17 then L={22,7,7,7,7,7,7,7,7,6,5}; if g==18 then L={19,6,6,6,6,6,6,6,5,5,5}; if g==19 then L={20,7,7,6,6,6,6,6,6,5,5}; if g==20 then L={20,7,6,6,6,6,6,6,6,6,5}; return L ) TEST /// linSysAlexander(16,17) try linSysAlexander(12,14) else "not implemented" /// properTransformAlexander=method(); -- Input: S, the coordinate ring of P4, L, a linear system on -- the Picard group of the Alexander surface Y obtained blowing -- up the points pts -- Output: an element of the linear system L, i.e., a curve in P4 properTransformAlexander(Ring, List, List) := (S, L, pts) -> ( if #L != 11 then error "Too few or too many coefficients"; Hs:=((gens truncate(13,intersect(apply(10,i->(pts_i)^4))))_{0..4}); I:=intersect(apply(10,i->(pts_i)^(L_(i+1)))); Es:=gens truncate (L_0,I); R:=ring Es; E:=ideal(Es*random(source Es,R^{-(L_0)})); RE:=R/E; return ker map(RE,S,sub(Hs,RE)) ) TEST /// (g,d)=(16,17) L=linSysAlexander(g,d) S = ZZ/32009[x_0..x_4]; (Y,pts):=alexanderSurface(S); I=properTransformAlexander(S,L,pts); codim I, genus I, degree I /// auxiliarLineBundle=method(TypicalValue=>Module); --- input: A and IE, ideals of a surface in P4 --- and of a curve on it --- output: a module on P4, restriction of Omega(1)_A on the --- curve IE auxiliarLineBundle(Ideal,Ideal):=(A,IE)->( S:=ring A; SE:=S/IE; omegaA:=Ext^2(S^1/A,S^{-5}); FomegaA:=res omegaA; M:=(FomegaA.dd_1)**(SE); L:=syz transpose syz transpose M; L1:=lift(L,S); return coker L1 ) TEST /// p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; H={13,4,4,4,4,4,4,4,4,4,4}; -- the hyperplane class on Y (Y,pts)=alexanderSurface(S); Z={31,10,10,10,10,10,10,9,9,9,9} time IZ=properTransformAlexander(S,Z,pts); -- takes about 26 sec N=auxiliarLineBundle(Y,IZ); betti res N /// ------------------------------------------------------------------ -- The functions for the construction of the curves -- ------------------------------------------------------------------ singularCurveInP4=method(TypicalValue=>Ideal) -- Input: the coordinate ring of P4 -- Output: a nodal curve of genus g and degree d in P4 singularCurveInP4(Ring, ZZ, ZZ) := (S,g,d) -> ( -- Construct a curve C of genus g with a D=g^1_(2g-2-d) and embed -- it in P4 via the complete linear system |K-D| k:=2*g-2-d; n:=char last S.baseRings-1; --(k,g,n):=(8,12,char last S.baseRings-1); --I:=idealOfNodalCurve(k,g,n); (P,Q,multL,f):=pickGoodPoints(k,g,n); K:=ideal sectionsFromPoints(P,Q); KminusD:=(gens intersect(K,ideal f_{0}))_{0..4}; phi:=map(ring P,S,KminusD); return ideal mingens ker phi ) TEST /// p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; time C=singularCurveInP4(S,10,12); codim C, genus C, degree C isSmoothCurve(C) time C=singularCurveInP4(S,11,13); codim C, genus C, degree C isSmoothCurve(C) time C=singularCurveInP4(S,12,14); codim C, genus C, degree C isSmoothCurve(C) time C=singularCurveInP4(S,13,15); codim C, genus C, degree C isSmoothCurve(C) /// curveGenus12Degree14InP4=method(TypicalValue=>Ideal) -- Input: the coordinate ring of P4 -- Output: a curve of genus 12 and degree 14 in P4 curveGenus12Degree14InP4(Ring) := (S) -> ( IC:=singularCurveInP4(S,12,14); omegaC:=Ext^2(IC,S^{-5}); -- canonical module of C fomega:=res omegaC; sM:=S^{-5}**coker transpose fomega.dd_3; psi:=first matrixFactorizationFromModule(sM); monadShape:=betti map(S^{2:-1}, S^{2:-1,2:-2},0); return idealFromMatFac(psi,monadShape) ) TEST /// p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; time C=curveGenus12Degree14InP4(S); codim C, genus C, degree C isSmoothCurve(C) /// curveGenus13Degree15InP4=method(TypicalValue=>Ideal) -- Input: the coordinate ring of P4 -- Output: a curve of genus 13 and degree 15 in P4 curveGenus13Degree15InP4(Ring) := (S) -> ( IC:=singularCurveInP4(S,13,15); omegaC:=Ext^2(IC,S^{-5}); -- canonical module of C fomega:=res omegaC; sM:=S^{-5}**coker transpose fomega.dd_3; psi:=first matrixFactorizationFromModule(sM); monadShape:=betti map(S^{3:-1}, S^{3:-1,2:-2},0); return idealFromMatFac(psi,monadShape) ) TEST /// p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; time C=curveGenus13Degree15InP4(S); codim C, genus C, degree C isSmoothCurve(C) /// randomCurveGenus10Degree13InP4=method(TypicalValue=>Ideal) -- Input: the coordinate ring of P4 -- Output: a curve of genus 10 and degree 13 in P4 randomCurveGenus10Degree13InP4(Ring) := (R) -> ( -- On P1xP2, we start with a rational curve of degree 4 together -- with 3 general lines. Call D''' their union. x:=symbol x; y:=symbol y; S:=coefficientRing(R)[x_0,x_1,y_0..y_2,Degrees=>{2:{1,0},3:{0,1}}]; -- Cox-ring of P^1 x P^2 m:=ideal basis({1,1},S); ICrat:=ideal random(S^1,S^{2:{-2,-1}}); ICratSat:=saturate(ICrat,m); ILines:=apply(3,i->ideal random(S^1,S^{{-1,0},{0,-1}})); IE'':=saturate(intersect(ICratSat,intersect ILines), ideal(x_0*y_0)); -- We choose random forms in IE'' of degree (4,2) that define -- the complete intersection X' and compute the ideal IE' of -- the linkage curve E' IX':=ideal(gens IE'' * random(source gens IE'',S^{2:{-4,-2}})); IE':=IX':IE''; IE'=saturate(IE',ideal(x_0*y_0)); -- We choose 5 random points on P1xP2 and choose random forms of -- degree (3,3) that define a complete intersection containing -- the 5 points; we compute the ideal IE of the linkage curve E fivePoints:=apply(5,i->ideal(random({1,0},S),random({0,1},S), random({0,1},S))); J:=intersect(fivePoints|{IE'}); genJ:=gens J; ci:=genJ_(select(rank source genJ,i-> degrees source genJ_{i}=={{3,3}})); IE:=saturate(ideal ci:IE',m); -- We compute the plane model of E and its embedding in -- P4 via the complete linear system K-P, where P is the divisor -- of the 5 points chosen previously. P2:=coefficientRing(R)[y_0..y_2]; pi2:=map(P2,S); IEPlane:=sub(IE,P2); RE:=P2/IEPlane; nodes:=sub(saturate ideal jacobian IE,P2); fivePoints2:=apply(fivePoints,p-> trim pi2 p); KminusP:=gens truncate(6,intersect(fivePoints2|{nodes})); emb:=map(RE,R,sub(KminusP,RE)); return kernel emb ) TEST /// p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; time C=randomCurveGenus10Degree13InP4(S); codim C, genus C, degree C isSmoothCurve(C) /// randomCurveGenus12Degree14InP4=method(TypicalValue=>Ideal) -- Input: the coordinate ring of P4 -- Output: a curve of genus 12 and degree 14 in P4 randomCurveGenus12Degree14InP4(Ring) := (S) -> ( IE:=randomCurveGenus10Degree13InP4(S); monadShape:=betti map(S^{2:-1}, S^{2:-1,2:-2},0); (phi,psi):=matrixFactorizationFromModule(IE); return idealFromMatFac(psi,monadShape) ) TEST /// p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; time C=randomCurveGenus12Degree14InP4(S); codim C, genus C, degree C isSmoothCurve(C) /// randomGenus12Degree8CoverOfP1=method() -- Input: the coordinate ring of P4 -- Output: a random canonical curve of genus 12 together -- with two hyperplanes in P^11 cutting out a g^1_8 randomGenus12Degree8CoverOfP1(Ring) := (R) -> ( Fp:=last R.baseRings; y:= symbol y; S:=Fp[y_0..y_4]; RS:= R**S; IC:=randomCurveGenus12Degree14InP4(S); omegaC:=presentation truncate(0,Ext^2(IC,S^{ -5})); H:=ideal random(S^1,S^{1: -1}); SH:=S/(H+IC); omegaCH:=presentation ((coker omegaC)**SH); graph:=substitute(vars R,RS)*substitute(omegaC,RS); graphH:=substitute(vars R,RS)*substitute(omegaCH,RS); linForm:=random(1,SH); J:=saturate(((ideal graph)),substitute(linForm,RS)); I:=ideal mingens substitute(J,R); JH:=saturate(((ideal graphH)),substitute(linForm,RS)); IH:=ideal mingens substitute(JH,R); H1:=(gens IH)_{0}; H2:=(gens IH)_{1}; return (I, H1, H2) ) TEST /// p=32009; Fp=ZZ/p; R=Fp[x_0..x_11]; time (I, H1, H2)=randomGenus12Degree8CoverOfP1(R); K=I+ideal(random(ZZ)*H1+random(ZZ)*H2); degree K, codim K IH=I+ideal(H1)+ideal(H2); degree IH, codim IH KmH=K:IH; degree KmH, codim KmH /// curveOnAlexanderSurface=method(); -- Input: the coordinate ring of P4 and a pair (g,d) -- Output: a curve of genus g and degree d, and an -- Alexander surface Y containing it curveOnAlexanderSurface(Ring,ZZ,ZZ) := (S,g,d) -> ( L:=linSysAlexander(g,d); (Y,pts):=alexanderSurface(S); return (properTransformAlexander(S,L,pts), Y) ) TEST /// p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; time alexC=first curveOnAlexanderSurface(S,16,17) codim C, genus C, degree C isSmoothCurve(C) try curveOnAlexanderSurface(S,15,15) /// ------------------------------------------------------------------ -- The functions needed to produce Betti candidates -- ------------------------------------------------------------------ isStrictlyIncreasing = method() -- input: list of rational numbers -- output true if the list is strictly increasing, else false. isStrictlyIncreasing(List) := L->( t:=true; for i from 0 to #L-2 do t=(t and (L_i( L:=lowestDegrees B; if not isStrictlyIncreasing(L) then return false; C:=pureBettiDiagram L; ratio:=min apply(#L, i->(B#(i,{L_i}, L_i))/(C#( i,{L_i},L_i))); (C,ratio,merge(B,C, (i,j)->i-ratio*j)) ) isInBoijSoederbergCone=method(TypicalValue=>Boolean) -- input: a BettiTally -- output: Boolean isInBoijSoederbergCone(BettiTally) := B-> ( Components:={}; B1:= new MutableHashTable from B; while min values B1 >= 0 and max values B1 > 0 do ( X:=decompose2(new BettiTally from B1); if class X===Boolean then return false; B1=new MutableHashTable from X_2; --change the type of the values in X_0 to ZZ Y:=new BettiTally from apply(pairs X_0, i->{ first i, lift(last i, ZZ)}); Components = append(Components, hold(X_1) * Y)); if min values B1 <0 then return false else return true; ) TEST /// B=new BettiTally from {(0, {1}, 1) => 6, (1, {2}, 2) => 10, (2, {3}, 3) => 3, (1, {3}, 3) => 3, (1, {4}, 4) => 1, (2, {5}, 5) => 13, (3, {6}, 6) => 9, (4, {7}, 7) => 1 } isInBoijSoederbergCone B B'=new BettiTally from {(0, {1}, 1) => 6, (1, {2}, 2) => 10, (2, {3}, 3) => 3, (1, {3}, 3) => 3, (1, {4}, 4) => 2, (2, {5}, 5) => 13, (3, {6}, 6) => 9, (4, {7}, 7) => 1 } isInBoijSoederbergCone B' /// bettiCandidates = method() -- input: beta, a BettiTally of a matrix factorization of a form -- of degree d, and d -- output: A List of possible Betti tables of modules in P4, lying -- in the Boij-Soederberg cone and whose support has -- codimension >= 3, which could lead to the desired -- matrix factorization without cancellations, with the -- additional assumption pd < 5 bettiCandidates(BettiTally,ZZ) := (beta,d) -> ( b01:=apply(keys beta,c->(c_0,c_2)); b0:=select(b01,c->c_0==0); b1:=select(b01,c->c_0==1); b2:=apply(b0,c->(c_0+2,c_1+d)); b3:=apply(b1,c->(c_0+2,c_1+d)); b4:=apply(b2,c->(c_0+2,c_1+d)); bs:=b0|b1|b2|b3|b4; bb:=symbol bb; Rb:=ZZ[apply(bs,c->bb_c)]; eqs:=apply(b0,c->sum(3,i->bb_(c_0+2*i,c_1+d*i))-beta_(( select(keys beta,cc->(cc_0,cc_2)==c))_0))| apply(b1,c->sum(2,i->bb_(c_0+2*i,c_1+d*i))-beta_(( select(keys beta,cc->(cc_0,cc_2)==c))_0)); -- equations given by the shape of the matrix factorization eqsc:={{sum apply(bs,c->(-1)^(c_0)*bb_c)}, {sum apply(bs,c->(-1)^(c_0)*(c_1)*bb_c)}, {sum apply(bs,c->(-1)^(c_0)*(c_1)*(c_1-1)*bb_c)}}; -- equations for codim >2 -- we select the equation among the ones involving b_(0,j) with -- the maximum coefficient. bounds:= apply(apply(#b0,k->eqs_k),eq->sub(eq,ZZ)); minBounds:= min bounds; toIso:=first first select(for i from 0 to #bounds-1 list { i,bounds_i},l->l_1==minBounds); eqsbut1:=(for j from 0 to toIso-1 list eqs_j) | ( for j from toIso+1 to #eqs-1 list eqs_j); -- we removed the equation selected above. As the solutions of the -- equations are natural numbers, one has to deal with partitions -- of the entries of the given shape of the matrix factorization. -- Any equation among the eqs is redundant because of the first -- equation in eqsc, hence removing the equation with the maximum -- coefficient will make the computations faster mapList:=apply(eqsbut1,eq->( coef:=coefficients eq; bound:=-sub(eq,ZZ); coef1:=(entries coef_0)_0; t:=#coef1-1; -- the length of the partition L:={{}}; b:=bound; while (#(L_0) < (t-1)) do ( L=flatten apply(#L,i->for k from 0 to b-sum(L_i) list ((L_i)|{k})) ); L=for i from 0 to #L-1 list ((L_i)|{b-sum(L_i)}); mapss2:=for i from 0 to #L-1 list ( ideal(for j from 0 to t-1 list (coef1_j) - (L_i)_j) ); mapss4:=apply(mapss2, c->vars Rb% c); mapss4)); ell:=#mapList-1; if ell==0 then mapList=flatten mapList; newLast:={}; while ell>0 do ( newLast=flatten apply(mapList_ell,m2->apply( mapList_(ell-1),m1->sub(m1,m2))); mapList=if ell>1 then (mapList_{0..ell-2}|{ newLast}) else newLast; ell=ell-1 ); -- now I have a list of partitions for all the coefficients -- appearing in the given shape, except the maximum one eqss:=apply(eqsc,c->{sub(c_0,mapList_0)}); A:=matrix (for j from 0 to 2 list ( noz:=(rank target (coefficients(matrix({ eqss_j})))_1)-1; if first flatten entries sub((matrix({ eqss_j})),ZZ) == 0 then noz=noz+1; apply(3-noz,i->0)|apply(noz,k -> (( coefficients(matrix({eqss_j})))_1)_(k,0)) )); -- the coefficients of the monomials appearing in each of eqsc -- depend only on the shape of the matrix factorization, so we -- compute them once and for all. They are encoded in A. A=sub(A,ZZ); L:={}; -- now, for every choice of multiple partitions in mapList, we -- look for natural solutions for the eqsc. Their coefficients -- in degree zero depend on the choice of the partitions, and are -- encoded in the matrix B. Remark: eqs_0 == eqsc_0. for i from 0 to #mapList-1 do ( eqss=apply(eqsc,c->{sub(c_0,mapList_i)}); B:=matrix for j from 0 to 2 list {-first flatten entries sub( (matrix({eqss_j})),ZZ)}; B=sub(B,ZZ); C:=B//A; if (all(flatten entries C,c-> c >=0) and (B-A*(C)==0)) then L=L|{{{i}}|{flatten entries C}}; ); freeVars:=((flatten entries (coefficients eqs_toIso)_0)_{0..2}); mapList1:=apply(L,c->sub(mapList_(c_0_0),apply( 3,x-> freeVars_x => (c_1)_x))); mapList=apply(mapList1,mapL->sub(mapL,ZZ)); -- now mapList encodes all the possible solutions to our system TList:={}; apply(mapList,mapL->( bsForBetti := apply(bs,ped->(first ped, { last ped}, last ped)); mapLEntries := flatten entries mapL; nonZero := select(#mapLEntries,i->mapLEntries_i>0); homDegreeOccurring := unique apply(apply( nonZero,i->bsForBetti_i),j->first j); if max homDegreeOccurring+1 == #homDegreeOccurring then TList = TList | {new BettiTally from apply( nonZero,i->bsForBetti_i => mapLEntries_i)}; )); -- we have constructed a list of tables, from which we select the -- ones in the Boij-Soederberg cone BScands:=select(TList,b->isInBoijSoederbergCone(b)); return BScands) TEST /// p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; beta=betti map(S^{9:0,1:-1},S^{1:-1,9:-2},0) d=3; time bettiCandidates(beta,d) /// -- This function is very close to the one above, but here we -- look for tables with pd = 5. This usually requires much longer -- time. bettiCandidatesIrrelevant = method() -- input: beta, a BettiTally of a matrix factorization of a form -- of degree d, and d -- output: A List of possible Betti tables of modules with the -- irrelevant ideal associated in P4, lying -- in the Boij-Soederberg cone and whose support has -- codimension >= 3, which could lead to the desired -- matrix factorization without cancellations bettiCandidatesIrrelevant(BettiTally,ZZ) := (beta,d) -> ( b01:=apply(keys beta,c->(c_0,c_2)); b0:=select(b01,c->c_0==0); b1:=select(b01,c->c_0==1); b2:=apply(b0,c->(c_0+2,c_1+d)); b3:=apply(b1,c->(c_0+2,c_1+d)); b4:=apply(b2,c->(c_0+2,c_1+d)); b5:=apply(b3,c->(c_0+2,c_1+d)); bs:=b0|b1|b2|b3|b4|b5; bb:=symbol bb; Rb:=ZZ[apply(bs,c->bb_c)]; eqs:=apply(b0,c->sum(3,i->bb_(c_0+2*i,c_1+d*i))-beta_(( select(keys beta,cc->(cc_0,cc_2)==c))_0))| apply(b1,c->sum(3,i->bb_(c_0+2*i,c_1+d*i))-beta_(( select(keys beta,cc->(cc_0,cc_2)==c))_0)); -- equations given by the shape of the matrix factorization eqsc:={{sum apply(bs,c->(-1)^(c_0)*bb_c)}, {sum apply(bs,c->(-1)^(c_0)*(c_1)*bb_c)}, {sum apply(bs,c->(-1)^(c_0)*(c_1)*(c_1-1)*bb_c)}}; -- equations for codim >2 -- we select the equation with the maximum coefficient among the -- equations involving the bb_(0,j) bounds:= apply(apply(#b0,k->eqs_k),eq->sub(eq,ZZ)); minBounds:= min bounds; toIso:=first first select(for i from 0 to #bounds-1 list { i,bounds_i},l->l_1==minBounds); eqsbut1:=(for j from 0 to toIso-1 list eqs_j) | ( for j from toIso+1 to #eqs-1 list eqs_j); -- we removed the equation selected above. As the solutions of the -- equations are natural numbers, one has to deal with partitions -- of the entries of the given shape of the matrix factorization. -- Any equation among the eqs is redundant because of the first -- equation in eqsc, hence removing the equation with the maximum -- coefficient will make the computations faster mapList:=apply(eqsbut1,eq->( coef:=coefficients eq; bound:=-sub(eq,ZZ); coef1:=(entries coef_0)_0; t:=#coef1-1; -- the length of the partition L:={{}}; b:=bound; while (#(L_0) < (t-1)) do ( L=flatten apply(#L,i->for k from 0 to b-sum( L_i) list ((L_i)|{k})) ); L=for i from 0 to #L-1 list ((L_i)|{b-sum(L_i)}); mapss2:=for i from 0 to #L-1 list ( ideal(for j from 0 to t-1 list ( coef1_j) - (L_i)_j) ); mapss4:=apply(mapss2, c->vars Rb% c); mapss4)); ell:=#mapList-1; if ell==0 then mapList=flatten mapList; newLast:={}; while ell>0 do ( newLast=flatten apply(mapList_ell,m2->apply( mapList_(ell-1),m1->sub(m1,m2))); mapList=if ell>1 then ( mapList_{0..ell-2}|{newLast}) else newLast; ell=ell-1 ); -- now I have a list of partitions for all the coefficients -- appearing in the given shape, except the maximum one among -- those appearing in the left column (target column) eqss:=apply(eqsc,c->{sub(c_0,mapList_0)}); A:=matrix (for j from 0 to 2 list ( noz:=(rank target (coefficients(matrix({eqss_j})))_1)-1; if first flatten entries sub((matrix( {eqss_j})),ZZ) == 0 then noz=noz+1; apply(3-noz,i->0)|apply(noz,k -> ((coefficients( matrix({eqss_j})))_1)_(k,0)) )); -- the coefficients of the monomials appearing in each of eqsc -- depend only on the shape of the matrix factorization, so we -- compute them once and for all. They are encoded in A. A=sub(A,ZZ); -- we select the only solutions for which the b_{5,j} will not be all zero, -- since we are focusing on the cases with the irrelevant ideal associated mapList=select(mapList,m->sum apply(#b5,i->first first entries (m)_(#bs-i-1)) != 0); L:={}; -- now, for every choice of multiple partitions in mapList, we -- look for natural solutions for the eqsc. Their coefficients -- in degree zero depend on the choice of the partitions, and are -- encoded in the matrix B. Remark: eqs_0 == eqsc_0. for i from 0 to #mapList-1 do ( eqss=apply(eqsc,c->{sub(c_0,mapList_i)}); B:=matrix for j from 0 to 2 list {-(first flatten entries sub((matrix({eqss_j})),ZZ))}; B=sub(B,ZZ); C:=B//A; if (all(flatten entries C,c-> c >=0) and (B-A*(C)==0)) then L=L|{{{i}}|{flatten entries C}}; ); freeVars:=((flatten entries (coefficients eqs_toIso)_0)_{0..2}); mapList1:=apply(L,c->sub(mapList_(c_0_0),apply( 3,x-> freeVars_x => (c_1)_x))); mapList=apply(mapList1,mapL->sub(mapL,ZZ)); -- now mapList encodes all the possible solutions to our system TList:={}; apply(mapList,mapL->( bsForBetti := apply(bs,ped->(first ped, { last ped}, last ped)); mapLEntries := flatten entries mapL; nonZero := select(#mapLEntries,i->mapLEntries_i>0); homDegreeOccurring := unique apply(apply( nonZero,i->bsForBetti_i),j->first j); if max homDegreeOccurring+1 == #homDegreeOccurring then TList = TList | {new BettiTally from apply( nonZero,i->bsForBetti_i => mapLEntries_i)}; )); -- we have constructed a list of tables, from which we select the -- ones in the Boij-Soederberg cone BScands:=select(TList,b->isInBoijSoederbergCone(b)); return BScands) TEST /// p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; beta=betti map(S^{9:0,1:-1},S^{1:-1,9:-2},0) d=3; time bettiCandidatesIrrelevant(beta,d) /// listOfPossibleMatFac=method() -- input: beta, a BettiTally of a matrix factorization of a -- form of degree d, and d -- output: a list of shapes of matrix factorizations, which give -- rise to a matrix factorization of the given shape up to -- taking syzygies, dual or both. listOfPossibleMatFac(BettiTally,ZZ):=(beta,d) -> ( x:= symbol x; S:=ZZ/101[x_0..x_4]; beta':=beta(last min keys beta); keysB:=keys beta'; beta1:=new BettiTally from apply(#keysB,i->(first keysB_i, { last keysB_i}, last keysB_i) => beta'_(keysB_i)); b01:=keys beta1; b0:=select(b01,c->c_0==0); b1:=select(b01,c->c_0==1); L0:=apply(b0,c->{beta1_c,c_2}); targ:=flatten apply(L0,l->for i from 1 to l_0 list -l_1); L1:=apply(b1,c->{beta1_c,c_2}); sour:=flatten apply(L1,l->for i from 1 to l_0 list -l_1); duals:=apply(targ,l->-l+min(targ|sour)); dualt:=apply(sour,l->-l+min(targ|sour)); beta2:=betti map(S^dualt,S^duals,0); syzt:=apply(targ,l->l-d-max(sour)); syzs:=apply(sour,l->l-max(sour)); beta3:=betti map(S^syzs,S^syzt,0); duals=apply(syzs,l->-l+min(syzs|syzt)); dualt=apply(syzt,l->-l+min(syzs|syzt)); beta4:=betti map(S^dualt,S^duals,0); return unique({beta1,beta2,beta3,beta4}) ) TEST /// p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; beta=betti map(S^{9:0,1:-1},S^{1:-1,9:-2},0) listOfPossibleMatFac(beta,3) listOfPossibleMatFac(beta,4) beta=betti map(S^{9:0,1:-1},S^{0:-1,10:-2},0) listOfPossibleMatFac(beta,3) /// candidateTables=method() -- input: beta, a BettiTally of a matrix factorization of a -- form of degree d, and d -- output: a list of candidates Betti tables whose induced matrix -- factorization has the fixed shape, assuming that no -- cancellation occurs, with pd < 5 candidateTables(BettiTally,ZZ) := (beta,d) -> ( L:=listOfPossibleMatFac(beta,d); cands:=flatten apply(L,l->bettiCandidates(l,d)); degs:=unique sort apply(cands,b->degree b); cands=flatten apply(degs,d->select(cands,b-> degree b==d)); return cands) TEST /// p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; beta=betti map(S^{9:0,1:-1},S^{1:-1,9:-2},0) d=3; time candidateTables(beta,d) /// candidateTablesWithIrrelevantIdealAssociated=method() -- input: beta, a BettiTally of a matrix factorization of a -- form of degree d, and d -- output: a list of candidates Betti tables whose induced matrix -- factorization has the fixed shape, assuming that no -- cancellation occurs, with pd = 5 candidateTablesWithIrrelevantIdealAssociated(BettiTally,ZZ) := (beta,d) -> ( L:=listOfPossibleMatFac(beta,d); cands:=flatten apply(L,l->bettiCandidatesIrrelevant(l,d)); degs:=unique sort apply(cands,b->degree b); cands=flatten apply(degs,d->select(cands,b-> degree b==d)); return cands) TEST /// p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; beta=betti map(S^{9:0,1:-1},S^{1:-1,9:-2},0) d=3; time candidateTablesWithIrrelevantIdealAssociated(beta,d) /// arisingMatFac=method() -- input: B, a BettiTally, and d, a natural number -- output: the shape of the matrix factorization of a form of -- degree d arising from B, assuming that all the -- cancellations that might occur will do. arisingMatFac(BettiTally,ZZ) := (B',d) -> ( B:= new BettiTally from apply (keys B',l->toSequence(toList(l)-{0,{last min keys B'},last min keys B'})=>B'_l); listOfRows0:=sort unique apply(select(keys B,C->even first C),C->{ 6,last C+d*floor((6-first C)/2)}); listOfRows1:=sort unique apply(select(keys B,C->odd first C),C->{ 5,last C+d*floor((5-first C)/2)}); listLength:=(last max listOfRows0)-6; C:=symbol C; for i from 0 to 7 do for j from 0 to listLength do C_(i,j)=B_(i,{i+j},i+j); -- C is my Betti table Macaulay2 style finC:=symbol finC; -- finC will be the final Betti table with all the cancellations shifted:=symbol shifted; leftCanc:=symbol leftCanc; rightCanc:=symbol rightCanc; provvC:= symbol provvC; for j from 0 to listLength do finC_(0,j)=C_(0,j); for j from 0 to listLength do provvC_(1,j)=C_(1,j); for i from 2 to 6 do ( for j from 0 to d-3 do shifted_(i,j)=0; for j from d-2 to listLength do shifted_(i,j)=finC_(i-2,j-d+2); for j from 0 to listLength-1 do rightCanc_j=min{shifted_(i,j),C_(i-1,j+1)}; rightCanc_(listLength)=0; leftCanc_0=0; for j from 1 to listLength do leftCanc_j=rightCanc_(j-1); for j from 0 to listLength do finC_(i-1,j)=provvC_(i-1,j)-leftCanc_j; for j from 0 to listLength do provvC_(i,j)=C_(i,j)-rightCanc_(j)+shifted_(i,j); ); -- after the execution of the i-th step, I have the (i-1)-th column. The i-th is -- temporary keysMatFac:={(for j from 0 to listLength list finC_(5,j))} | {(for j from 0 to listLength list provvC_(6,j))} ; nonZeroFirst:=positions(first keysMatFac,n->n>0); nonZeroSecond:=positions(last keysMatFac,n->n>0); shift:=first nonZeroFirst; return new BettiTally from apply(nonZeroFirst,i->(0,{i-shift},i-shift)=>(first keysMatFac)_i) | apply(nonZeroSecond,i->(1,{i+1-shift},i+1-shift)=>(last keysMatFac)_i) ) TEST /// p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; beta=betti map(S^{12:0,1:-1},S^{1:-1,12:-2},0) d=3; L=candidateTables(beta,3) apply(L,l->arisingMatFac(l,d)) /// precompiledListOfCandidates=method() precompiledListOfCandidates(ZZ) := (g) -> ( if g == 16 then return {new BettiTally from {(0,{0},0) => 1, (1,{3},3) => 11, (2,{4},4) => 14, (2,{6},6) => 4, --------------------------------------------------------------------------------------------- (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 8, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 3, (1,{1},1) => 1, (1,{3},3) => 14, (2,{4},4) => 15, (2,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 5, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 4, (1,{1},1) => 1, (1,{2},2) => 1, (1,{3},3) => 14, (2,{4},4) => 15, (3,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{7},7) => 5, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (1,{3},3) => 11, --------------------------------------------------------------------------------------------- (2,{4},4) => 14, (2,{5},5) => 3, (2,{6},6) => 1, (3,{5},5) => 4, (3,{7},7) => 8, (4,{8},8) => --------------------------------------------------------------------------------------------- 4}, new BettiTally from {(0,{1},1) => 2, (1,{3},3) => 13, (2,{4},4) => 16, (2,{5},5) => 1, --------------------------------------------------------------------------------------------- (2,{6},6) => 1, (3,{5},5) => 4, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 4, (1,{3},3) => 7, (1,{4},4) => 1, (2,{6},6) => 14, (3,{7},7) => 12, (3,{8},8) --------------------------------------------------------------------------------------------- => 2, (3,{9},9) => 1, (4,{8},8) => 4, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 7, (1,{4},4) => 3, (2,{4},4) => 2, (2,{6},6) => 14, (3,{7},7) => 12, (3,{9},9) --------------------------------------------------------------------------------------------- => 1, (4,{8},8) => 2, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 13, (2,{4},4) => 15, (2,{6},6) => 3, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 6, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 4, (1,{1},1) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 16, (2,{4},4) => 16, (3,{6},6) => 3, (3,{7},7) => 3, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 4, (1,{2},2) => 3, (1,{3},3) => 13, (2,{4},4) --------------------------------------------------------------------------------------------- => 15, (3,{5},5) => 1, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{3},3) => 13, (2,{4},4) => 15, (2,{5},5) => 3, (3,{5},5) => 4, (3,{7},7) --------------------------------------------------------------------------------------------- => 6, (4,{8},8) => 3}, new BettiTally from {(0,{1},1) => 2, (0,{2},2) => 1, (1,{3},3) => 15, --------------------------------------------------------------------------------------------- (2,{4},4) => 17, (2,{5},5) => 1, (3,{5},5) => 4, (3,{7},7) => 4, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 4, (1,{3},3) => 7, (1,{5},5) => 4, (2,{5},5) => 2, (2,{6},6) => --------------------------------------------------------------------------------------------- 15, (3,{7},7) => 12, (4,{8},8) => 1, (4,{9},9) => 1}, new BettiTally from {(0,{2},2) => 5, --------------------------------------------------------------------------------------------- (1,{3},3) => 9, (1,{4},4) => 3, (2,{4},4) => 3, (2,{6},6) => 13, (3,{7},7) => 10, (3,{9},9) --------------------------------------------------------------------------------------------- => 1, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 3, (1,{3},3) => 4, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 1, (2,{6},6) => 16, (3,{7},7) => 15, (4,{8},8) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 15, (2,{4},4) => 16, (2,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 4, (1,{2},2) => 2, (1,{3},3) => 15, (2,{4},4) => 16, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 1, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 4, (1,{5},5) => 4, (2,{5},5) => 1, (2,{6},6) => 19, (3,{7},7) => 15, (4,{9},9) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 1, (1,{3},3) => 6, (1,{5},5) => 4, --------------------------------------------------------------------------------------------- (2,{5},5) => 3, (2,{6},6) => 17, (3,{7},7) => 13, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 5, (1,{3},3) => 9, (1,{5},5) => 4, (2,{4},4) => 1, (2,{5},5) => 2, (2,{6},6) => --------------------------------------------------------------------------------------------- 14, (3,{7},7) => 10, (4,{9},9) => 1}, new BettiTally from {(0,{2},2) => 6, (1,{3},3) => 11, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (2,{4},4) => 4, (2,{6},6) => 12, (3,{7},7) => 8, (3,{9},9) => 1, (4,{10},10) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{2},2) => 4, (1,{3},3) => 6, (1,{4},4) => 2, (1,{5},5) => 1, --------------------------------------------------------------------------------------------- (2,{6},6) => 15, (3,{7},7) => 13, (3,{8},8) => 1, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 4, (1,{3},3) => 6, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 1, (2,{6},6) => --------------------------------------------------------------------------------------------- 15, (3,{7},7) => 13, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 17, (2,{4},4) => 17, (2,{6},6) => 1, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 2, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, (1,{1},1) => 1, (1,{3},3) => 12, --------------------------------------------------------------------------------------------- (2,{4},4) => 14, (2,{6},6) => 4, (3,{6},6) => 3, (3,{7},7) => 7, (4,{8},8) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 4, (1,{2},2) => 1, (1,{3},3) => 17, (2,{4},4) --------------------------------------------------------------------------------------------- => 17, (3,{5},5) => 1, (3,{6},6) => 2, (3,{7},7) => 2, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{1},1) => 1, (1,{3},3) => 14, (2,{4},4) => 16, (2,{5},5) => 2, (2,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{5},5) => 4, (3,{7},7) => 5, (4,{8},8) => 2}, new BettiTally from {(0,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 16, (2,{4},4) => 18, (2,{6},6) => 1, (3,{5},5) => 4, (3,{7},7) => 3, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 8, (1,{4},4) => 3, (1,{5},5) => 1, --------------------------------------------------------------------------------------------- (2,{4},4) => 2, (2,{6},6) => 14, (3,{7},7) => 11, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 4, (1,{3},3) => 19, (2,{4},4) => 18, (3,{5},5) => 1, (3,{6},6) --------------------------------------------------------------------------------------------- => 3}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 1, (1,{1},1) => 1, (1,{3},3) => 14, --------------------------------------------------------------------------------------------- (2,{4},4) => 15, (2,{6},6) => 3, (3,{6},6) => 3, (3,{7},7) => 5, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 4, (1,{1},1) => 1, (1,{2},2) => 3, (1,{3},3) => --------------------------------------------------------------------------------------------- 14, (2,{4},4) => 15, (3,{7},7) => 5, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{1},1) => 1, (0,{2},2) => 1, (1,{3},3) => 16, (2,{4},4) => 17, (2,{5},5) => 2, (3,{5},5) --------------------------------------------------------------------------------------------- => 4, (3,{7},7) => 3, (4,{8},8) => 1}, new BettiTally from {(0,{1},1) => 3, (0,{2},2) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 18, (2,{4},4) => 19, (3,{5},5) => 4, (3,{7},7) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 6, (1,{3},3) => 10, (1,{5},5) => 4, (2,{5},5) => 3, (2,{6},6) => 13, (3,{7},7) --------------------------------------------------------------------------------------------- => 9, (4,{8},8) => 1}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 7, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (2,{6},6) => 14, (3,{7},7) => 12, (3,{9},9) => 1, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 6, (1,{3},3) => 10, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 3, (2,{6},6) --------------------------------------------------------------------------------------------- => 13, (3,{7},7) => 9, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 2, --------------------------------------------------------------------------------------------- (1,{1},1) => 1, (1,{3},3) => 16, (2,{4},4) => 16, (2,{6},6) => 2, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 3, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 4, (1,{1},1) => 1, --------------------------------------------------------------------------------------------- (1,{2},2) => 2, (1,{3},3) => 16, (2,{4},4) => 16, (3,{6},6) => 1, (3,{7},7) => 3, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 7, (1,{5},5) => 4, --------------------------------------------------------------------------------------------- (2,{5},5) => 2, (2,{6},6) => 17, (3,{7},7) => 12, (4,{9},9) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 7, (1,{3},3) => 12, (1,{5},5) => 4, (2,{4},4) => 1, (2,{5},5) => 3, (2,{6},6) --------------------------------------------------------------------------------------------- => 12, (3,{7},7) => 7}, new BettiTally from {(0,{2},2) => 6, (1,{3},3) => 9, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (2,{4},4) => 1, (2,{6},6) => 13, (3,{7},7) => 10, (3,{9},9) => 1, (4,{8},8) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 7, (1,{3},3) => 12, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) --------------------------------------------------------------------------------------------- => 4, (2,{6},6) => 12, (3,{7},7) => 7}, new BettiTally from {(0,{0},0) => 2, (1,{3},3) => 15, --------------------------------------------------------------------------------------------- (2,{4},4) => 16, (2,{6},6) => 4, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 4, (4,{8},8) => --------------------------------------------------------------------------------------------- 1}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 3, (1,{1},1) => 1, (1,{3},3) => 18, --------------------------------------------------------------------------------------------- (2,{4},4) => 17, (2,{6},6) => 1, (3,{6},6) => 3, (3,{7},7) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 4, (1,{1},1) => 1, (1,{2},2) => 1, (1,{3},3) => 18, (2,{4},4) --------------------------------------------------------------------------------------------- => 17, (3,{6},6) => 2, (3,{7},7) => 1}, new BettiTally from {(0,{0},0) => 2, (1,{3},3) => 15, --------------------------------------------------------------------------------------------- (2,{4},4) => 16, (2,{5},5) => 3, (2,{6},6) => 1, (3,{5},5) => 4, (3,{7},7) => 4, (4,{8},8) => --------------------------------------------------------------------------------------------- 1}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 2, (1,{3},3) => 17, (2,{4},4) => 18, --------------------------------------------------------------------------------------------- (2,{5},5) => 1, (2,{6},6) => 1, (3,{5},5) => 4, (3,{7},7) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{1},1) => 3, (1,{1},1) => 1, (1,{3},3) => 17, (2,{4},4) => 18, (2,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{5},5) => 3, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{1},1) => 3, (1,{3},3) => 15, (2,{4},4) => 16, (2,{6},6) => 1, (3,{5},5) => 1, (3,{7},7) --------------------------------------------------------------------------------------------- => 4, (4,{8},8) => 1}, new BettiTally from {(0,{2},2) => 7, (1,{3},3) => 11, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (2,{4},4) => 2, (2,{6},6) => 12, (3,{7},7) => 8, (3,{9},9) => 1, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 1, (1,{3},3) => 17, (2,{4},4) => 17, (2,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 4, (1,{2},2) => 3, (1,{3},3) => 17, (2,{4},4) => 17, (3,{5},5) => 1, (3,{7},7) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 1, (1,{3},3) => 17, (2,{4},4) => 17, --------------------------------------------------------------------------------------------- (2,{5},5) => 3, (3,{5},5) => 4, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 3, (1,{3},3) => 17, (2,{4},4) => 17, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{7},7) => 2}, new BettiTally from {(0,{2},2) => 8, (1,{3},3) => 13, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (2,{4},4) => 3, (2,{6},6) => 11, (3,{7},7) => 6, (3,{9},9) => 1, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 4, (1,{3},3) => 10, (1,{5},5) => 4, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 15, (3,{7},7) => 9}, new BettiTally from {(0,{2},2) => 9, (1,{3},3) => 15, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (2,{4},4) => 4, (2,{6},6) => 10, (3,{7},7) => 4, (3,{9},9) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 3, (1,{1},1) => 1, (1,{3},3) => 16, (2,{4},4) => 16, (2,{6},6) --------------------------------------------------------------------------------------------- => 4, (3,{6},6) => 3, (3,{7},7) => 3}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{1},1) => 4, (1,{3},3) => 16, (2,{4},4) => 16, (2,{6},6) => 1, (3,{7},7) => 3}}; ---------------------- ---------------------- ---------------------- if g == 17 then return {new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 1, (1,{3},3) => 14, (2,{4},4) => 17, --------------------------------------------------------------------------------------------- (2,{6},6) => 4, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 8, (4,{8},8) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 4, (1,{1},1) => 1, (1,{3},3) => 17, (2,{4},4) --------------------------------------------------------------------------------------------- => 18, (2,{6},6) => 1, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 5, (4,{8},8) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 1, (1,{1},1) => 2, (1,{3},3) => 12, (2,{4},4) --------------------------------------------------------------------------------------------- => 15, (2,{6},6) => 4, (3,{6},6) => 3, (3,{7},7) => 10, (4,{8},8) => 5}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 5, (1,{1},1) => 1, (1,{2},2) => 1, (1,{3},3) => 17, (2,{4},4) --------------------------------------------------------------------------------------------- => 18, (3,{5},5) => 1, (3,{6},6) => 2, (3,{7},7) => 5, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 1, (1,{3},3) => 14, (2,{4},4) => 17, (2,{5},5) => 3, (2,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{5},5) => 5, (3,{7},7) => 8, (4,{8},8) => 4}, new BettiTally from {(0,{1},1) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{3},3) => 16, (2,{4},4) => 19, (2,{5},5) => 1, (2,{6},6) => 1, (3,{5},5) --------------------------------------------------------------------------------------------- => 5, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (1,{3},3) => 5, --------------------------------------------------------------------------------------------- (1,{5},5) => 5, (2,{4},4) => 1, (2,{5},5) => 2, (2,{6},6) => 21, (3,{7},7) => 17, (4,{9},9) --------------------------------------------------------------------------------------------- => 1, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 4, (1,{3},3) => 8, (1,{5},5) => 5, --------------------------------------------------------------------------------------------- (2,{4},4) => 2, (2,{5},5) => 1, (2,{6},6) => 18, (3,{7},7) => 14, (4,{9},9) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 5, (1,{3},3) => 10, (1,{5},5) => 5, (2,{4},4) => 2, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 16, (3,{7},7) => 12, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 6, (1,{3},3) => 12, (1,{4},4) => 3, (2,{4},4) => 5, (2,{6},6) => 14, (3,{7},7) => 10, --------------------------------------------------------------------------------------------- (3,{9},9) => 2, (4,{10},10) => 2}, new BettiTally from {(0,{2},2) => 4, (1,{3},3) => 7, --------------------------------------------------------------------------------------------- (1,{4},4) => 1, (1,{5},5) => 1, (2,{6},6) => 17, (3,{7},7) => 15, (3,{8},8) => 2, (3,{9},9) --------------------------------------------------------------------------------------------- => 1, (4,{8},8) => 5, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 4, (1,{3},3) => 7, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 2, (2,{6},6) => 17, (3,{7},7) => 15, (3,{9},9) --------------------------------------------------------------------------------------------- => 1, (4,{8},8) => 3, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => --------------------------------------------------------------------------------------------- 10, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 5, (2,{6},6) => 16, (3,{7},7) => 12, --------------------------------------------------------------------------------------------- (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 16, --------------------------------------------------------------------------------------------- (2,{4},4) => 18, (2,{6},6) => 3, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 6, (4,{8},8) => --------------------------------------------------------------------------------------------- 3}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 5, (1,{1},1) => 1, (1,{3},3) => 19, --------------------------------------------------------------------------------------------- (2,{4},4) => 19, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 3, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 2, (1,{1},1) => 2, (1,{3},3) => 14, (2,{4},4) --------------------------------------------------------------------------------------------- => 16, (2,{6},6) => 3, (3,{6},6) => 3, (3,{7},7) => 8, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 5, (1,{2},2) => 3, (1,{3},3) => 16, (2,{4},4) => 18, (3,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 5, --------------------------------------------------------------------------------------------- (1,{1},1) => 2, (1,{2},2) => 3, (1,{3},3) => 14, (2,{4},4) => 16, (3,{7},7) => 8, (4,{8},8) --------------------------------------------------------------------------------------------- => 4}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 16, (2,{4},4) => 18, --------------------------------------------------------------------------------------------- (2,{5},5) => 3, (3,{5},5) => 5, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{1},1) => 2, (0,{2},2) => 2, (1,{3},3) => 18, (2,{4},4) => 20, (2,{5},5) => 1, (3,{5},5) --------------------------------------------------------------------------------------------- => 5, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from {(0,{2},2) => 4, (1,{3},3) => 7, --------------------------------------------------------------------------------------------- (1,{5},5) => 5, (2,{5},5) => 2, (2,{6},6) => 18, (3,{7},7) => 15, (4,{8},8) => 2, (4,{9},9) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{2},2) => 4, (1,{3},3) => 6, (1,{4},4) => 3, (2,{6},6) => 17, --------------------------------------------------------------------------------------------- (3,{7},7) => 16, (3,{9},9) => 2, (4,{8},8) => 5, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 5, (1,{3},3) => 9, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 3, (2,{6},6) => --------------------------------------------------------------------------------------------- 16, (3,{7},7) => 13, (3,{9},9) => 1, (4,{8},8) => 2, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 3, (1,{3},3) => 4, (1,{4},4) => 3, (1,{5},5) => 2, (2,{6},6) => 19, (3,{7},7) --------------------------------------------------------------------------------------------- => 18, (4,{8},8) => 5}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 3, (1,{3},3) => 18, --------------------------------------------------------------------------------------------- (2,{4},4) => 19, (2,{6},6) => 2, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 4, (4,{8},8) => --------------------------------------------------------------------------------------------- 2}, new BettiTally from {(0,{0},0) => 2, (1,{1},1) => 1, (1,{3},3) => 13, (2,{4},4) => 16, --------------------------------------------------------------------------------------------- (2,{6},6) => 5, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 9, (4,{8},8) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 3, (1,{1},1) => 2, (1,{3},3) => 16, (2,{4},4) --------------------------------------------------------------------------------------------- => 17, (2,{6},6) => 2, (3,{6},6) => 3, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 5, (1,{2},2) => 2, (1,{3},3) => 18, (2,{4},4) => 19, (3,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{6},6) => 1, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 5, (1,{1},1) => 2, (1,{2},2) => 2, (1,{3},3) => 16, (2,{4},4) => 17, (3,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 15, (2,{4},4) => 18, (2,{5},5) => 2, (2,{6},6) => 2, (3,{5},5) => 5, (3,{7},7) --------------------------------------------------------------------------------------------- => 7, (4,{8},8) => 3}, new BettiTally from {(0,{1},1) => 3, (1,{3},3) => 17, (2,{4},4) => 20, --------------------------------------------------------------------------------------------- (2,{6},6) => 2, (3,{5},5) => 5, (3,{7},7) => 5, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{1},1) => 3, (1,{1},1) => 5, (1,{3},3) => 16, (2,{4},4) => 19, (2,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{7},7) => 6, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 1, (1,{3},3) => 4, --------------------------------------------------------------------------------------------- (1,{5},5) => 5, (2,{5},5) => 1, (2,{6},6) => 22, (3,{7},7) => 18, (4,{8},8) => 1, (4,{9},9) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 1, (1,{3},3) => 6, (1,{5},5) => 5, --------------------------------------------------------------------------------------------- (2,{5},5) => 3, (2,{6},6) => 20, (3,{7},7) => 16, (4,{8},8) => 1, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 5, (1,{3},3) => 9, (1,{5},5) => 5, (2,{4},4) => 1, (2,{5},5) => --------------------------------------------------------------------------------------------- 2, (2,{6},6) => 17, (3,{7},7) => 13, (4,{8},8) => 1, (4,{9},9) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 5, (1,{3},3) => 8, (1,{4},4) => 3, (2,{4},4) => 1, (2,{6},6) => 16, (3,{7},7) --------------------------------------------------------------------------------------------- => 14, (3,{9},9) => 2, (4,{8},8) => 4, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 6, (1,{3},3) => 11, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 4, (2,{6},6) => 15, --------------------------------------------------------------------------------------------- (3,{7},7) => 11, (3,{9},9) => 1, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 4, (1,{3},3) => 6, (1,{4},4) => 2, (1,{5},5) => 2, (2,{6},6) => 18, (3,{7},7) --------------------------------------------------------------------------------------------- => 16, (3,{8},8) => 1, (4,{8},8) => 5}, new BettiTally from {(0,{2},2) => 4, (1,{3},3) => 6, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 1, (2,{6},6) => 18, (3,{7},7) => 16, (4,{8},8) --------------------------------------------------------------------------------------------- => 4}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 4, (1,{3},3) => 20, (2,{4},4) => 20, --------------------------------------------------------------------------------------------- (2,{6},6) => 1, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 2, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 1, (1,{1},1) => 1, (1,{3},3) => 15, (2,{4},4) --------------------------------------------------------------------------------------------- => 17, (2,{6},6) => 4, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 7, (4,{8},8) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 4, (1,{1},1) => 2, (1,{3},3) => 18, (2,{4},4) --------------------------------------------------------------------------------------------- => 18, (2,{6},6) => 1, (3,{6},6) => 3, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 5, (1,{2},2) => 1, (1,{3},3) => 20, (2,{4},4) => 20, (3,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{6},6) => 2, (3,{7},7) => 2, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 5, (1,{1},1) => 2, (1,{2},2) => 1, (1,{3},3) => 18, (2,{4},4) => 18, (3,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{3},3) => 17, (2,{4},4) => 19, (2,{5},5) => 2, (2,{6},6) => 1, (3,{5},5) --------------------------------------------------------------------------------------------- => 5, (3,{7},7) => 5, (4,{8},8) => 2}, new BettiTally from {(0,{1},1) => 3, (0,{2},2) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 19, (2,{4},4) => 21, (2,{6},6) => 1, (3,{5},5) => 5, (3,{7},7) => 3, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 1, (1,{3},3) => 6, (1,{5},5) => 5, --------------------------------------------------------------------------------------------- (2,{4},4) => 1, (2,{5},5) => 1, (2,{6},6) => 21, (3,{7},7) => 16, (4,{9},9) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 8, (1,{5},5) => 5, (2,{4},4) => --------------------------------------------------------------------------------------------- 1, (2,{5},5) => 3, (2,{6},6) => 19, (3,{7},7) => 14, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 6, (1,{3},3) => 11, (1,{5},5) => 5, (2,{4},4) => 2, (2,{5},5) => 2, (2,{6},6) --------------------------------------------------------------------------------------------- => 16, (3,{7},7) => 11, (4,{9},9) => 1}, new BettiTally from {(0,{2},2) => 6, (1,{3},3) => --------------------------------------------------------------------------------------------- 10, (1,{4},4) => 3, (2,{4},4) => 2, (2,{6},6) => 15, (3,{7},7) => 12, (3,{9},9) => 2, --------------------------------------------------------------------------------------------- (4,{8},8) => 3, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 7, (1,{3},3) => 13, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 5, (2,{6},6) => 14, (3,{7},7) => 9, (3,{9},9) => --------------------------------------------------------------------------------------------- 1, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 8, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (1,{5},5) => 2, (2,{4},4) => 2, (2,{6},6) => 17, (3,{7},7) => 14, (4,{8},8) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 5, (1,{3},3) => 22, (2,{4},4) => 21, (3,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{6},6) => 3}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 2, (1,{1},1) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 17, (2,{4},4) => 18, (2,{6},6) => 3, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 5, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 5, (1,{1},1) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 20, (2,{4},4) => 19, (3,{6},6) => 3, (3,{7},7) => 2, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 5, (1,{1},1) => 1, (1,{2},2) => 3, (1,{3},3) => --------------------------------------------------------------------------------------------- 17, (2,{4},4) => 18, (3,{5},5) => 1, (3,{7},7) => 5, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{1},1) => 1, (0,{2},2) => 2, (1,{3},3) => 19, (2,{4},4) => 20, (2,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{5},5) => 5, (3,{7},7) => 3, (4,{8},8) => 1}, new BettiTally from {(0,{1},1) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 2, (1,{3},3) => 21, (2,{4},4) => 22, (3,{5},5) => 5, (3,{7},7) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 6, (1,{3},3) => 10, (1,{5},5) => 5, (2,{5},5) => 3, (2,{6},6) --------------------------------------------------------------------------------------------- => 16, (3,{7},7) => 12, (4,{8},8) => 2}, new BettiTally from {(0,{2},2) => 7, (1,{3},3) => --------------------------------------------------------------------------------------------- 12, (1,{4},4) => 3, (2,{4},4) => 3, (2,{6},6) => 14, (3,{7},7) => 10, (3,{9},9) => 2, --------------------------------------------------------------------------------------------- (4,{8},8) => 2, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 7, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 1, (2,{6},6) => 17, (3,{7},7) => 15, (3,{9},9) => 1, (4,{8},8) --------------------------------------------------------------------------------------------- => 5}, new BettiTally from {(0,{2},2) => 6, (1,{3},3) => 10, (1,{4},4) => 3, (1,{5},5) => 2, --------------------------------------------------------------------------------------------- (2,{4},4) => 3, (2,{6},6) => 16, (3,{7},7) => 12, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (1,{3},3) => 16, (2,{4},4) => 18, (2,{6},6) => 5, (3,{5},5) => 2, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 6, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 3, --------------------------------------------------------------------------------------------- (1,{1},1) => 1, (1,{3},3) => 19, (2,{4},4) => 19, (2,{6},6) => 2, (3,{5},5) => 1, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 3, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, (1,{1},1) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 14, (2,{4},4) => 16, (2,{6},6) => 5, (3,{6},6) => 3, (3,{7},7) => 8, (4,{8},8) --------------------------------------------------------------------------------------------- => 3}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 5, (1,{1},1) => 1, (1,{2},2) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 19, (2,{4},4) => 19, (3,{5},5) => 1, (3,{6},6) => 1, (3,{7},7) => 3, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 2, (1,{3},3) => 16, (2,{4},4) => 18, (2,{5},5) => 3, --------------------------------------------------------------------------------------------- (2,{6},6) => 2, (3,{5},5) => 5, (3,{7},7) => 6, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{1},1) => 2, (1,{3},3) => 18, (2,{4},4) => 20, (2,{5},5) => 1, (2,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{5},5) => 5, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (1,{1},1) => 1, (1,{3},3) => 18, (2,{4},4) => 20, (2,{6},6) => 2, (3,{5},5) --------------------------------------------------------------------------------------------- => 4, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{1},1) => 3, (1,{3},3) => 16, (2,{4},4) => 18, (2,{6},6) => 2, (3,{5},5) => 2, (3,{7},7) --------------------------------------------------------------------------------------------- => 6, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (1,{1},1) => 5, --------------------------------------------------------------------------------------------- (1,{3},3) => 14, (2,{4},4) => 16, (2,{6},6) => 2, (3,{7},7) => 8, (4,{8},8) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 7, (1,{5},5) => 5, (2,{5},5) => --------------------------------------------------------------------------------------------- 2, (2,{6},6) => 20, (3,{7},7) => 15, (4,{8},8) => 1, (4,{9},9) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 7, (1,{3},3) => 12, (1,{5},5) => 5, (2,{4},4) => 1, (2,{5},5) => 3, (2,{6},6) --------------------------------------------------------------------------------------------- => 15, (3,{7},7) => 10, (4,{8},8) => 1}, new BettiTally from {(0,{2},2) => 8, (1,{3},3) => --------------------------------------------------------------------------------------------- 14, (1,{4},4) => 3, (2,{4},4) => 4, (2,{6},6) => 13, (3,{7},7) => 8, (3,{9},9) => 2, --------------------------------------------------------------------------------------------- (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 6, (1,{3},3) => 9, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 1, (2,{6},6) => 16, (3,{7},7) => 13, (3,{9},9) --------------------------------------------------------------------------------------------- => 1, (4,{8},8) => 4}, new BettiTally from {(0,{2},2) => 7, (1,{3},3) => 12, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (1,{5},5) => 2, (2,{4},4) => 4, (2,{6},6) => 15, (3,{7},7) => 10, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 1, (1,{3},3) => 18, (2,{4},4) => 19, (2,{6},6) --------------------------------------------------------------------------------------------- => 4, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 4, (1,{1},1) => 1, (1,{3},3) => 21, (2,{4},4) => 20, (2,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 1}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{1},1) => 2, (1,{3},3) => 16, (2,{4},4) => 17, (2,{6},6) => 4, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 6, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 5, --------------------------------------------------------------------------------------------- (1,{1},1) => 1, (1,{2},2) => 1, (1,{3},3) => 21, (2,{4},4) => 20, (3,{5},5) => 1, (3,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{7},7) => 1}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 1, (1,{3},3) => 18, --------------------------------------------------------------------------------------------- (2,{4},4) => 19, (2,{5},5) => 3, (2,{6},6) => 1, (3,{5},5) => 5, (3,{7},7) => 4, (4,{8},8) => --------------------------------------------------------------------------------------------- 1}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 2, (0,{2},2) => 1, (1,{3},3) => 20, --------------------------------------------------------------------------------------------- (2,{4},4) => 21, (2,{5},5) => 1, (2,{6},6) => 1, (3,{5},5) => 5, (3,{7},7) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 1, (1,{3},3) => --------------------------------------------------------------------------------------------- 20, (2,{4},4) => 21, (2,{6},6) => 1, (3,{5},5) => 4, (3,{7},7) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 3, (1,{3},3) => 18, (2,{4},4) --------------------------------------------------------------------------------------------- => 19, (2,{6},6) => 1, (3,{5},5) => 2, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 5, (1,{3},3) => 16, (2,{4},4) --------------------------------------------------------------------------------------------- => 17, (2,{6},6) => 1, (3,{7},7) => 6, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{3},3) => 9, (1,{5},5) => 5, (2,{4},4) => 1, (2,{5},5) => 2, (2,{6},6) => --------------------------------------------------------------------------------------------- 19, (3,{7},7) => 13, (4,{9},9) => 1}, new BettiTally from {(0,{2},2) => 8, (1,{3},3) => 14, --------------------------------------------------------------------------------------------- (1,{5},5) => 5, (2,{4},4) => 2, (2,{5},5) => 3, (2,{6},6) => 14, (3,{7},7) => 8}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 9, (1,{3},3) => 16, (1,{4},4) => 3, (2,{4},4) => 5, (2,{6},6) --------------------------------------------------------------------------------------------- => 12, (3,{7},7) => 6, (3,{9},9) => 2, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 7, (1,{3},3) => 11, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 2, (2,{6},6) => 15, --------------------------------------------------------------------------------------------- (3,{7},7) => 11, (3,{9},9) => 1, (4,{8},8) => 3}, new BettiTally from {(0,{2},2) => 8, --------------------------------------------------------------------------------------------- (1,{3},3) => 14, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 5, (2,{6},6) => 14, (3,{7},7) --------------------------------------------------------------------------------------------- => 8}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 2, (1,{3},3) => 20, (2,{4},4) => 20, --------------------------------------------------------------------------------------------- (2,{6},6) => 3, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{2},2) => 2, (1,{1},1) => 2, (1,{3},3) => 18, (2,{4},4) => 18, (2,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{6},6) => 3, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 5, (1,{2},2) => 3, (1,{3},3) => 20, (2,{4},4) => 20, (3,{5},5) => 2, (3,{7},7) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 5, (1,{1},1) => 2, (1,{2},2) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 18, (2,{4},4) => 18, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 2, (1,{3},3) => 20, (2,{4},4) => 20, (2,{5},5) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 5, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, (0,{2},2) => 2, --------------------------------------------------------------------------------------------- (1,{1},1) => 3, (1,{3},3) => 20, (2,{4},4) => 20, (3,{5},5) => 2, (3,{7},7) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 8, (1,{3},3) => 13, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 14, (3,{7},7) => 9, (3,{9},9) => 1, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (1,{1},1) => 1, (1,{3},3) => 17, (2,{4},4) => 18, (2,{6},6) => 5, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 5, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{1},1) => 2, (1,{3},3) => 20, (2,{4},4) => 19, (2,{6},6) => 2, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 5, (1,{1},1) => 2, --------------------------------------------------------------------------------------------- (1,{2},2) => 2, (1,{3},3) => 20, (2,{4},4) => 19, (3,{6},6) => 1, (3,{7},7) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{1},1) => 1, (1,{3},3) => 19, (2,{4},4) => 20, (2,{5},5) --------------------------------------------------------------------------------------------- => 2, (2,{6},6) => 2, (3,{5},5) => 5, (3,{7},7) => 3}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (1,{1},1) => 2, (1,{3},3) => 19, (2,{4},4) => 20, (2,{6},6) => 2, (3,{5},5) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 3}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (1,{1},1) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 17, (2,{4},4) => 18, (2,{6},6) => 2, (3,{5},5) => 1, (3,{7},7) => 5, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 4, (1,{3},3) => 10, (1,{5},5) => 5, --------------------------------------------------------------------------------------------- (2,{5},5) => 3, (2,{6},6) => 18, (3,{7},7) => 12, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 9, (1,{3},3) => 15, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 4, (2,{6},6) --------------------------------------------------------------------------------------------- => 13, (3,{7},7) => 7, (3,{9},9) => 1, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{1},1) => 1, (1,{3},3) => 19, (2,{4},4) => 19, (2,{6},6) => 4, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 3}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{1},1) => 4, (1,{3},3) => 19, (2,{4},4) => 19, (2,{6},6) => 1, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{7},7) => 3}, new BettiTally from {(0,{0},0) => 2, (1,{3},3) => 7, (1,{5},5) => 5, --------------------------------------------------------------------------------------------- (2,{5},5) => 2, (2,{6},6) => 22, (3,{7},7) => 15, (4,{9},9) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 5, (1,{3},3) => 12, (1,{5},5) => 5, (2,{4},4) => 1, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 17, (3,{7},7) => 10}, new BettiTally from {(0,{2},2) => 10, (1,{3},3) => --------------------------------------------------------------------------------------------- 17, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 5, (2,{6},6) => 12, (3,{7},7) => 5, --------------------------------------------------------------------------------------------- (3,{9},9) => 1}, new BettiTally from {(0,{2},2) => 10, (1,{3},3) => 16, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (2,{4},4) => 3, (2,{6},6) => 12, (3,{7},7) => 6, (3,{9},9) => 2, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 4, (1,{1},1) => 2, (1,{3},3) => 18, (2,{4},4) => 18, (2,{6},6) --------------------------------------------------------------------------------------------- => 5, (3,{6},6) => 3, (3,{7},7) => 4}, new BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{1},1) => 5, (1,{3},3) => 18, (2,{4},4) => 18, (2,{6},6) => 2, (3,{7},7) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 11, (1,{3},3) => 18, (1,{4},4) => 3, (2,{4},4) => 4, (2,{6},6) --------------------------------------------------------------------------------------------- => 11, (3,{7},7) => 4, (3,{9},9) => 2, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 2, (1,{3},3) => 10, (1,{5},5) => 5, (2,{5},5) => 3, (2,{6},6) => 20, (3,{7},7) --------------------------------------------------------------------------------------------- => 12}, new BettiTally from {(0,{2},2) => 12, (1,{3},3) => 20, (1,{4},4) => 3, (2,{4},4) => --------------------------------------------------------------------------------------------- 5, (2,{6},6) => 10, (3,{7},7) => 2, (3,{9},9) => 2}}; ---------------------- if g == 18 then return {new BettiTally from {(0,{0},0) => 1, (1,{3},3) => 5, (1,{5},5) => 6, (2,{4},4) => 1, --------------------------------------------------------------------------------------------- (2,{5},5) => 2, (2,{6},6) => 24, (3,{7},7) => 20, (4,{8},8) => 1, (4,{9},9) => 1, (4,{10},10) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{2},2) => 4, (1,{3},3) => 8, (1,{5},5) => 6, (2,{4},4) => 2, --------------------------------------------------------------------------------------------- (2,{5},5) => 1, (2,{6},6) => 21, (3,{7},7) => 17, (4,{8},8) => 1, (4,{9},9) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 5, (1,{3},3) => 10, (1,{5},5) => 6, (2,{4},4) => 2, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 19, (3,{7},7) => 15, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally --------------------------------------------------------------------------------------------- from {(0,{2},2) => 5, (1,{3},3) => 9, (1,{4},4) => 3, (2,{4},4) => 2, (2,{6},6) => 18, --------------------------------------------------------------------------------------------- (3,{7},7) => 16, (3,{9},9) => 3, (4,{8},8) => 4, (4,{10},10) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 6, (1,{3},3) => 12, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 5, (2,{6},6) --------------------------------------------------------------------------------------------- => 17, (3,{7},7) => 13, (3,{9},9) => 2, (4,{8},8) => 1, (4,{10},10) => 2}, new BettiTally --------------------------------------------------------------------------------------------- from {(0,{2},2) => 4, (1,{3},3) => 7, (1,{4},4) => 1, (1,{5},5) => 2, (2,{6},6) => 20, --------------------------------------------------------------------------------------------- (3,{7},7) => 18, (3,{8},8) => 2, (3,{9},9) => 1, (4,{8},8) => 6, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 4, (1,{3},3) => 7, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => --------------------------------------------------------------------------------------------- 2, (2,{6},6) => 20, (3,{7},7) => 18, (3,{9},9) => 1, (4,{8},8) => 4, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 5, (1,{3},3) => 10, (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) --------------------------------------------------------------------------------------------- => 5, (2,{6},6) => 19, (3,{7},7) => 15, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally --------------------------------------------------------------------------------------------- from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 17, (2,{4},4) => 20, (2,{6},6) => 4, --------------------------------------------------------------------------------------------- (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) => 8, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 5, (1,{1},1) => 1, (1,{3},3) => 20, (2,{4},4) => 21, (2,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 5, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 2, (1,{1},1) => 2, (1,{3},3) => 15, (2,{4},4) => 18, (2,{6},6) --------------------------------------------------------------------------------------------- => 4, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 10, (4,{8},8) => 5}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 5, (1,{1},1) => 3, (1,{3},3) => 18, (2,{4},4) => 19, (2,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 7, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 6, (1,{1},1) => 1, (1,{2},2) => 1, (1,{3},3) => 20, (2,{4},4) => 21, (3,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{6},6) => 2, (3,{7},7) => 5, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 6, (1,{1},1) => 3, (1,{2},2) => 1, (1,{3},3) => 18, (2,{4},4) => 19, (3,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{7},7) => 7, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 17, (2,{4},4) => 20, (2,{5},5) => 3, (2,{6},6) => 1, (3,{5},5) => 6, (3,{7},7) --------------------------------------------------------------------------------------------- => 8, (4,{8},8) => 4}, new BettiTally from {(0,{1},1) => 2, (0,{2},2) => 2, (1,{3},3) => 19, --------------------------------------------------------------------------------------------- (2,{4},4) => 22, (2,{5},5) => 1, (2,{6},6) => 1, (3,{5},5) => 6, (3,{7},7) => 6, (4,{8},8) => --------------------------------------------------------------------------------------------- 3}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 1, (1,{3},3) => 7, (1,{5},5) => 4, --------------------------------------------------------------------------------------------- (2,{4},4) => 2, (2,{6},6) => 23, (3,{7},7) => 18, (3,{9},9) => 2, (4,{9},9) => 3, (4,{10},10) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 1, (1,{3},3) => 5, (1,{5},5) => 6, (2,{4},4) => 2, --------------------------------------------------------------------------------------------- (2,{6},6) => 25, (3,{7},7) => 20, (4,{9},9) => 3}, new BettiTally from {(0,{2},2) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 7, (1,{5},5) => 6, (2,{5},5) => 2, (2,{6},6) => 21, (3,{7},7) => 18, (4,{8},8) --------------------------------------------------------------------------------------------- => 3, (4,{9},9) => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 1, (1,{3},3) => 7, --------------------------------------------------------------------------------------------- (1,{5},5) => 6, (2,{4},4) => 2, (2,{5},5) => 2, (2,{6},6) => 23, (3,{7},7) => 18, (4,{9},9) --------------------------------------------------------------------------------------------- => 1, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 10, (1,{5},5) => --------------------------------------------------------------------------------------------- 6, (2,{4},4) => 3, (2,{5},5) => 1, (2,{6},6) => 20, (3,{7},7) => 15, (4,{9},9) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 6, (1,{3},3) => 12, (1,{5},5) => 6, (2,{4},4) => 3, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 18, (3,{7},7) => 13, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 6, (1,{3},3) => 11, (1,{4},4) => 3, (2,{4},4) => 3, (2,{6},6) => 17, (3,{7},7) => 14, --------------------------------------------------------------------------------------------- (3,{9},9) => 3, (4,{8},8) => 3, (4,{10},10) => 2}, new BettiTally from {(0,{2},2) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 6, (1,{4},4) => 3, (1,{5},5) => 1, (2,{6},6) => 20, (3,{7},7) => 19, (3,{9},9) --------------------------------------------------------------------------------------------- => 2, (4,{8},8) => 6, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 7, (1,{3},3) => --------------------------------------------------------------------------------------------- 14, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 6, (2,{6},6) => 16, (3,{7},7) => 11, --------------------------------------------------------------------------------------------- (3,{9},9) => 2, (4,{10},10) => 2}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 9, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 3, (2,{6},6) => 19, (3,{7},7) => 16, (3,{9},9) --------------------------------------------------------------------------------------------- => 1, (4,{8},8) => 3, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 3, (1,{3},3) => 4, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 3, (2,{6},6) => 22, (3,{7},7) => 21, (4,{8},8) => 6}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 6, (1,{3},3) => 12, (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) --------------------------------------------------------------------------------------------- => 6, (2,{6},6) => 18, (3,{7},7) => 13, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => --------------------------------------------------------------------------------------------- 1, (0,{2},2) => 3, (1,{3},3) => 19, (2,{4},4) => 21, (2,{6},6) => 3, (3,{5},5) => 3, --------------------------------------------------------------------------------------------- (3,{6},6) => 3, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (1,{1},1) => 1, (1,{3},3) => 14, (2,{4},4) => 18, (2,{6},6) => 6, (3,{5},5) => 2, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 11, (4,{8},8) => 5}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 6, --------------------------------------------------------------------------------------------- (1,{1},1) => 1, (1,{3},3) => 22, (2,{4},4) => 22, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 3, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 3, (1,{1},1) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 17, (2,{4},4) => 19, (2,{6},6) => 3, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 8, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 3, (1,{1},1) => 3, (1,{3},3) => 12, --------------------------------------------------------------------------------------------- (2,{4},4) => 16, (2,{6},6) => 6, (3,{6},6) => 3, (3,{7},7) => 13, (4,{8},8) => 6}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 6, (1,{1},1) => 3, (1,{3},3) => 20, (2,{4},4) --------------------------------------------------------------------------------------------- => 20, (3,{6},6) => 3, (3,{7},7) => 5, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 6, (1,{2},2) => 3, (1,{3},3) => 19, (2,{4},4) => 21, (3,{5},5) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 6, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 6, (1,{1},1) => 2, --------------------------------------------------------------------------------------------- (1,{2},2) => 3, (1,{3},3) => 17, (2,{4},4) => 19, (3,{5},5) => 1, (3,{7},7) => 8, (4,{8},8) --------------------------------------------------------------------------------------------- => 4}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 1, (1,{3},3) => 16, (2,{4},4) => 20, --------------------------------------------------------------------------------------------- (2,{5},5) => 2, (2,{6},6) => 3, (3,{5},5) => 6, (3,{7},7) => 9, (4,{8},8) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{1},1) => 3, (1,{3},3) => 18, (2,{4},4) => 22, (2,{6},6) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 6, (3,{7},7) => 7, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, (1,{3},3) => 19, --------------------------------------------------------------------------------------------- (2,{4},4) => 23, (2,{5},5) => 3, (2,{6},6) => 2, (3,{5},5) => 6, (3,{7},7) => 6, (4,{10},10) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 3, (1,{3},3) => 19, (2,{4},4) => 21, --------------------------------------------------------------------------------------------- (2,{5},5) => 3, (3,{5},5) => 6, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{1},1) => 2, (0,{2},2) => 3, (1,{3},3) => 21, (2,{4},4) => 23, (2,{5},5) => 1, (3,{5},5) --------------------------------------------------------------------------------------------- => 6, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{1},1) => 3, (1,{3},3) => 19, (2,{4},4) => 23, (2,{6},6) => 2, (3,{5},5) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 6, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (1,{1},1) => 5, --------------------------------------------------------------------------------------------- (1,{3},3) => 17, (2,{4},4) => 21, (2,{6},6) => 2, (3,{5},5) => 1, (3,{7},7) => 8, (4,{8},8) --------------------------------------------------------------------------------------------- => 1, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 1, (1,{3},3) => 4, (1,{5},5) => 6, --------------------------------------------------------------------------------------------- (2,{5},5) => 1, (2,{6},6) => 25, (3,{7},7) => 21, (4,{8},8) => 2, (4,{9},9) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 1, (1,{3},3) => 6, (1,{5},5) => 6, (2,{5},5) => --------------------------------------------------------------------------------------------- 3, (2,{6},6) => 23, (3,{7},7) => 19, (4,{8},8) => 2, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 5, (1,{3},3) => 9, (1,{5},5) => 6, (2,{4},4) => 1, (2,{5},5) => 2, (2,{6},6) => --------------------------------------------------------------------------------------------- 20, (3,{7},7) => 16, (4,{8},8) => 2, (4,{9},9) => 1}, new BettiTally from {(0,{2},2) => 7, --------------------------------------------------------------------------------------------- (1,{3},3) => 13, (1,{4},4) => 3, (2,{4},4) => 4, (2,{6},6) => 16, (3,{7},7) => 12, (3,{9},9) --------------------------------------------------------------------------------------------- => 3, (4,{8},8) => 2, (4,{10},10) => 2}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 8, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 1, (2,{6},6) => 19, (3,{7},7) => 17, (3,{9},9) --------------------------------------------------------------------------------------------- => 2, (4,{8},8) => 5, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 6, (1,{3},3) => --------------------------------------------------------------------------------------------- 11, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 4, (2,{6},6) => 18, (3,{7},7) => 14, --------------------------------------------------------------------------------------------- (3,{9},9) => 1, (4,{8},8) => 2, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 6, (1,{4},4) => 2, (1,{5},5) => 3, (2,{6},6) => 21, (3,{7},7) => 19, (3,{8},8) --------------------------------------------------------------------------------------------- => 1, (4,{8},8) => 6}, new BettiTally from {(0,{2},2) => 4, (1,{3},3) => 6, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (1,{5},5) => 3, (2,{4},4) => 1, (2,{6},6) => 21, (3,{7},7) => 19, (4,{8},8) => 5}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 4, (1,{3},3) => 21, (2,{4},4) => 22, (2,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 1, (1,{1},1) => 1, (1,{3},3) => 16, (2,{4},4) => 19, (2,{6},6) --------------------------------------------------------------------------------------------- => 5, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 9, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 4, (1,{1},1) => 2, (1,{3},3) => 19, (2,{4},4) => 20, (2,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{2},2) => 1, (1,{1},1) => 3, (1,{3},3) => 14, (2,{4},4) => 17, (2,{6},6) --------------------------------------------------------------------------------------------- => 5, (3,{6},6) => 3, (3,{7},7) => 11, (4,{8},8) => 5}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 6, (1,{2},2) => 2, (1,{3},3) => 21, (2,{4},4) => 22, (3,{5},5) => 3, (3,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 6, --------------------------------------------------------------------------------------------- (1,{1},1) => 2, (1,{2},2) => 2, (1,{3},3) => 19, (2,{4},4) => 20, (3,{5},5) => 1, (3,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{3},3) => 18, (2,{4},4) => 21, (2,{5},5) => 2, (2,{6},6) => 2, (3,{5},5) --------------------------------------------------------------------------------------------- => 6, (3,{7},7) => 7, (4,{8},8) => 3}, new BettiTally from {(0,{1},1) => 3, (0,{2},2) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 20, (2,{4},4) => 23, (2,{6},6) => 2, (3,{5},5) => 6, (3,{7},7) => 5, (4,{8},8) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 5, --------------------------------------------------------------------------------------------- (1,{3},3) => 19, (2,{4},4) => 22, (2,{6},6) => 1, (3,{5},5) => 1, (3,{7},7) => 6, (4,{10},10) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 1, (1,{3},3) => 6, (1,{5},5) => 6, --------------------------------------------------------------------------------------------- (2,{4},4) => 1, (2,{5},5) => 1, (2,{6},6) => 24, (3,{7},7) => 19, (4,{8},8) => 1, (4,{9},9) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 8, (1,{5},5) => 6, --------------------------------------------------------------------------------------------- (2,{4},4) => 1, (2,{5},5) => 3, (2,{6},6) => 22, (3,{7},7) => 17, (4,{8},8) => 1, (4,{10},10) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{2},2) => 6, (1,{3},3) => 11, (1,{5},5) => 6, (2,{4},4) => 2, --------------------------------------------------------------------------------------------- (2,{5},5) => 2, (2,{6},6) => 19, (3,{7},7) => 14, (4,{8},8) => 1, (4,{9},9) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 8, (1,{3},3) => 15, (1,{4},4) => 3, (2,{4},4) => 5, (2,{6},6) --------------------------------------------------------------------------------------------- => 15, (3,{7},7) => 10, (3,{9},9) => 3, (4,{8},8) => 1, (4,{10},10) => 2}, new BettiTally --------------------------------------------------------------------------------------------- from {(0,{2},2) => 6, (1,{3},3) => 10, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 2, --------------------------------------------------------------------------------------------- (2,{6},6) => 18, (3,{7},7) => 15, (3,{9},9) => 2, (4,{8},8) => 4, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 7, (1,{3},3) => 13, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) --------------------------------------------------------------------------------------------- => 5, (2,{6},6) => 17, (3,{7},7) => 12, (3,{9},9) => 1, (4,{8},8) => 1, (4,{10},10) => 1}, --------------------------------------------------------------------------------------------- new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 8, (1,{4},4) => 3, (1,{5},5) => 3, --------------------------------------------------------------------------------------------- (2,{4},4) => 2, (2,{6},6) => 20, (3,{7},7) => 17, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 5, (1,{3},3) => 23, (2,{4},4) => 23, (2,{6},6) => 1, (3,{5},5) --------------------------------------------------------------------------------------------- => 3, (3,{6},6) => 3, (3,{7},7) => 2, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 2, (1,{1},1) => 1, (1,{3},3) => 18, (2,{4},4) => 20, (2,{6},6) => 4, (3,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{6},6) => 3, (3,{7},7) => 7, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 5, (1,{1},1) => 2, (1,{3},3) => 21, (2,{4},4) => 21, (2,{6},6) => 1, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 2, (1,{1},1) => 3, (1,{3},3) => 16, (2,{4},4) => 18, (2,{6},6) => 4, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 9, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 6, --------------------------------------------------------------------------------------------- (1,{2},2) => 1, (1,{3},3) => 23, (2,{4},4) => 23, (3,{5},5) => 3, (3,{6},6) => 2, (3,{7},7) --------------------------------------------------------------------------------------------- => 2, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 6, (1,{1},1) => 2, --------------------------------------------------------------------------------------------- (1,{2},2) => 1, (1,{3},3) => 21, (2,{4},4) => 21, (3,{5},5) => 1, (3,{6},6) => 2, (3,{7},7) --------------------------------------------------------------------------------------------- => 4, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 1, (0,{2},2) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 20, (2,{4},4) => 22, (2,{5},5) => 2, (2,{6},6) => 1, (3,{5},5) => 6, (3,{7},7) --------------------------------------------------------------------------------------------- => 5, (4,{8},8) => 2}, new BettiTally from {(0,{1},1) => 3, (0,{2},2) => 2, (1,{3},3) => 22, --------------------------------------------------------------------------------------------- (2,{4},4) => 24, (2,{6},6) => 1, (3,{5},5) => 6, (3,{7},7) => 3, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 8, (1,{5},5) => 5, (2,{4},4) => --------------------------------------------------------------------------------------------- 2, (2,{6},6) => 23, (3,{7},7) => 17, (3,{9},9) => 1, (4,{9},9) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 8, (1,{5},5) => 6, (2,{4},4) => 2, (2,{5},5) => --------------------------------------------------------------------------------------------- 1, (2,{6},6) => 23, (3,{7},7) => 17, (4,{9},9) => 2}, new BettiTally from {(0,{2},2) => 6, --------------------------------------------------------------------------------------------- (1,{3},3) => 10, (1,{5},5) => 6, (2,{5},5) => 3, (2,{6},6) => 19, (3,{7},7) => 15, (4,{8},8) --------------------------------------------------------------------------------------------- => 3}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 3, (1,{3},3) => 10, (1,{5},5) => 6, --------------------------------------------------------------------------------------------- (2,{4},4) => 2, (2,{5},5) => 3, (2,{6},6) => 21, (3,{7},7) => 15, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 7, (1,{3},3) => 13, (1,{5},5) => 6, (2,{4},4) => 3, (2,{5},5) --------------------------------------------------------------------------------------------- => 2, (2,{6},6) => 18, (3,{7},7) => 12, (4,{9},9) => 1}, new BettiTally from {(0,{2},2) => 9, --------------------------------------------------------------------------------------------- (1,{3},3) => 17, (1,{4},4) => 3, (2,{4},4) => 6, (2,{6},6) => 14, (3,{7},7) => 8, (3,{9},9) --------------------------------------------------------------------------------------------- => 3, (4,{10},10) => 2}, new BettiTally from {(0,{2},2) => 7, (1,{3},3) => 12, (1,{4},4) => --------------------------------------------------------------------------------------------- 3, (1,{5},5) => 1, (2,{4},4) => 3, (2,{6},6) => 17, (3,{7},7) => 13, (3,{9},9) => 2, --------------------------------------------------------------------------------------------- (4,{8},8) => 3, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 7, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 2, (2,{6},6) => 20, (3,{7},7) => 18, (3,{9},9) => 1, (4,{8},8) --------------------------------------------------------------------------------------------- => 6}, new BettiTally from {(0,{2},2) => 8, (1,{3},3) => 15, (1,{4},4) => 3, (1,{5},5) => 2, --------------------------------------------------------------------------------------------- (2,{4},4) => 6, (2,{6},6) => 16, (3,{7},7) => 10, (3,{9},9) => 1, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 6, (1,{3},3) => 10, (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 19, (3,{7},7) => 15, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 17, (2,{4},4) => 20, (2,{6},6) => 6, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 8, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 6, (1,{3},3) => 25, --------------------------------------------------------------------------------------------- (2,{4},4) => 24, (3,{5},5) => 3, (3,{6},6) => 3}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{1},1) => 1, (1,{3},3) => 20, (2,{4},4) => 21, (2,{6},6) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{6},6) => 3, (3,{7},7) => 5, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (1,{1},1) => 2, (1,{3},3) => 15, (2,{4},4) => 18, (2,{6},6) => 6, (3,{5},5) => 1, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 10, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 6, --------------------------------------------------------------------------------------------- (1,{1},1) => 2, (1,{3},3) => 23, (2,{4},4) => 22, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 2, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 3, (1,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 18, (2,{4},4) => 19, (2,{6},6) => 3, (3,{6},6) => 3, (3,{7},7) => 7, (4,{8},8) --------------------------------------------------------------------------------------------- => 3}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 6, (1,{1},1) => 1, (1,{2},2) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 20, (2,{4},4) => 21, (3,{5},5) => 2, (3,{7},7) => 5, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 3, (0,{2},2) => 6, (1,{1},1) => 3, (1,{2},2) => 3, (1,{3},3) => --------------------------------------------------------------------------------------------- 18, (2,{4},4) => 19, (3,{7},7) => 7, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 17, (2,{4},4) => 20, (2,{5},5) => 3, (2,{6},6) => 3, (3,{5},5) => 6, (3,{7},7) --------------------------------------------------------------------------------------------- => 8, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 2, (1,{3},3) => 19, --------------------------------------------------------------------------------------------- (2,{4},4) => 22, (2,{5},5) => 1, (2,{6},6) => 3, (3,{5},5) => 6, (3,{7},7) => 6, (4,{8},8) => --------------------------------------------------------------------------------------------- 2}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 1, (0,{2},2) => 3, (1,{3},3) => 22, --------------------------------------------------------------------------------------------- (2,{4},4) => 23, (2,{5},5) => 2, (3,{5},5) => 6, (3,{7},7) => 3, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{1},1) => 3, (0,{2},2) => 3, (1,{3},3) => 24, (2,{4},4) => 25, (3,{5},5) --------------------------------------------------------------------------------------------- => 6, (3,{7},7) => 1}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 3, (1,{1},1) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 19, (2,{4},4) => 22, (2,{6},6) => 3, (3,{5},5) => 5, (3,{7},7) => 6, (4,{8},8) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, (1,{1},1) => 3, (1,{3},3) => 17, --------------------------------------------------------------------------------------------- (2,{4},4) => 20, (2,{6},6) => 3, (3,{5},5) => 3, (3,{7},7) => 8, (4,{8},8) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (1,{1},1) => 5, (1,{3},3) => 15, (2,{4},4) --------------------------------------------------------------------------------------------- => 18, (2,{6},6) => 3, (3,{5},5) => 1, (3,{7},7) => 10, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 4, (0,{1},1) => 3, (1,{1},1) => 6, (1,{3},3) => 18, (2,{4},4) => 21, (2,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{7},7) => 7, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 7, (1,{5},5) => 6, (2,{5},5) => 2, (2,{6},6) => 23, (3,{7},7) => 18, (4,{8},8) --------------------------------------------------------------------------------------------- => 2, (4,{9},9) => 1}, new BettiTally from {(0,{2},2) => 7, (1,{3},3) => 12, (1,{5},5) => 6, --------------------------------------------------------------------------------------------- (2,{4},4) => 1, (2,{5},5) => 3, (2,{6},6) => 18, (3,{7},7) => 13, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 8, (1,{3},3) => 14, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) --------------------------------------------------------------------------------------------- => 4, (2,{6},6) => 16, (3,{7},7) => 11, (3,{9},9) => 2, (4,{8},8) => 2, (4,{10},10) => 1}, --------------------------------------------------------------------------------------------- new BettiTally from {(0,{2},2) => 6, (1,{3},3) => 9, (1,{4},4) => 3, (1,{5},5) => 2, --------------------------------------------------------------------------------------------- (2,{4},4) => 1, (2,{6},6) => 19, (3,{7},7) => 16, (3,{9},9) => 1, (4,{8},8) => 5}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 7, (1,{3},3) => 12, (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) --------------------------------------------------------------------------------------------- => 4, (2,{6},6) => 18, (3,{7},7) => 13, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{3},3) => 19, (2,{4},4) => 21, (2,{6},6) => 5, (3,{5},5) => 3, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 6, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 4, --------------------------------------------------------------------------------------------- (1,{1},1) => 1, (1,{3},3) => 22, (2,{4},4) => 22, (2,{6},6) => 2, (3,{5},5) => 2, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 3, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 1, --------------------------------------------------------------------------------------------- (1,{1},1) => 2, (1,{3},3) => 17, (2,{4},4) => 19, (2,{6},6) => 5, (3,{5},5) => 1, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 8, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 4, --------------------------------------------------------------------------------------------- (1,{1},1) => 3, (1,{3},3) => 20, (2,{4},4) => 20, (2,{6},6) => 2, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 5, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 6, (1,{1},1) => 1, --------------------------------------------------------------------------------------------- (1,{2},2) => 2, (1,{3},3) => 22, (2,{4},4) => 22, (3,{5},5) => 2, (3,{6},6) => 1, (3,{7},7) --------------------------------------------------------------------------------------------- => 3, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 6, (1,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{2},2) => 2, (1,{3},3) => 20, (2,{4},4) => 20, (3,{6},6) => 1, (3,{7},7) => 5, (4,{8},8) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 1, (1,{3},3) => 19, (2,{4},4) => 21, --------------------------------------------------------------------------------------------- (2,{5},5) => 3, (2,{6},6) => 2, (3,{5},5) => 6, (3,{7},7) => 6, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{1},1) => 2, (0,{2},2) => 1, (1,{3},3) => 21, (2,{4},4) --------------------------------------------------------------------------------------------- => 23, (2,{5},5) => 1, (2,{6},6) => 2, (3,{5},5) => 6, (3,{7},7) => 4, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 1, (1,{3},3) => --------------------------------------------------------------------------------------------- 21, (2,{4},4) => 23, (2,{6},6) => 2, (3,{5},5) => 5, (3,{7},7) => 4, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 3, (1,{3},3) => --------------------------------------------------------------------------------------------- 19, (2,{4},4) => 21, (2,{6},6) => 2, (3,{5},5) => 3, (3,{7},7) => 6, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 5, (1,{3},3) => --------------------------------------------------------------------------------------------- 17, (2,{4},4) => 19, (2,{6},6) => 2, (3,{5},5) => 1, (3,{7},7) => 8, (4,{8},8) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 3, (1,{3},3) => 9, (1,{5},5) => 6, (2,{4},4) => --------------------------------------------------------------------------------------------- 1, (2,{5},5) => 2, (2,{6},6) => 22, (3,{7},7) => 16, (4,{8},8) => 1, (4,{9},9) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 8, (1,{3},3) => 14, (1,{5},5) => 6, (2,{4},4) => 2, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 17, (3,{7},7) => 11, (4,{8},8) => 1}, new BettiTally from {(0,{2},2) => 8, --------------------------------------------------------------------------------------------- (1,{3},3) => 13, (1,{4},4) => 3, (2,{4},4) => 2, (2,{6},6) => 16, (3,{7},7) => 12, (3,{9},9) --------------------------------------------------------------------------------------------- => 3, (4,{8},8) => 4, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 9, (1,{3},3) => --------------------------------------------------------------------------------------------- 16, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 5, (2,{6},6) => 15, (3,{7},7) => 9, --------------------------------------------------------------------------------------------- (3,{9},9) => 2, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 7, --------------------------------------------------------------------------------------------- (1,{3},3) => 11, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 2, (2,{6},6) => 18, (3,{7},7) --------------------------------------------------------------------------------------------- => 14, (3,{9},9) => 1, (4,{8},8) => 4}, new BettiTally from {(0,{2},2) => 8, (1,{3},3) => 14, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) => 5, (2,{6},6) => 17, (3,{7},7) => 11, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 2, (1,{3},3) => 21, (2,{4},4) => 22, --------------------------------------------------------------------------------------------- (2,{6},6) => 4, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) => 4, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 5, (1,{1},1) => 1, (1,{3},3) => 24, (2,{4},4) --------------------------------------------------------------------------------------------- => 23, (2,{6},6) => 1, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{2},2) => 2, (1,{1},1) => 2, (1,{3},3) => 19, (2,{4},4) => 20, (2,{6},6) --------------------------------------------------------------------------------------------- => 4, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 6, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{2},2) => 5, (1,{1},1) => 3, (1,{3},3) => 22, (2,{4},4) => 21, (2,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 3, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 6, (1,{1},1) => 1, (1,{2},2) => 1, (1,{3},3) => 24, (2,{4},4) => 23, (3,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{6},6) => 2, (3,{7},7) => 1}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 6, --------------------------------------------------------------------------------------------- (1,{1},1) => 3, (1,{2},2) => 1, (1,{3},3) => 22, (2,{4},4) => 21, (3,{6},6) => 2, (3,{7},7) --------------------------------------------------------------------------------------------- => 3, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 2, (1,{3},3) => 21, --------------------------------------------------------------------------------------------- (2,{4},4) => 22, (2,{5},5) => 3, (2,{6},6) => 1, (3,{5},5) => 6, (3,{7},7) => 4, (4,{8},8) => --------------------------------------------------------------------------------------------- 1}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 2, (0,{2},2) => 2, (1,{3},3) => 23, --------------------------------------------------------------------------------------------- (2,{4},4) => 24, (2,{5},5) => 1, (2,{6},6) => 1, (3,{5},5) => 6, (3,{7},7) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 1, (1,{3},3) => --------------------------------------------------------------------------------------------- 23, (2,{4},4) => 24, (2,{6},6) => 1, (3,{5},5) => 5, (3,{7},7) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 3, (1,{3},3) => 21, (2,{4},4) --------------------------------------------------------------------------------------------- => 22, (2,{6},6) => 1, (3,{5},5) => 3, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 5, (1,{3},3) => 19, (2,{4},4) --------------------------------------------------------------------------------------------- => 20, (2,{6},6) => 1, (3,{5},5) => 1, (3,{7},7) => 6, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (1,{3},3) => 8, (1,{5},5) => 6, (2,{4},4) => 1, (2,{5},5) => 3, (2,{6},6) => --------------------------------------------------------------------------------------------- 24, (3,{7},7) => 17, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 11, (1,{5},5) => 6, (2,{4},4) => 2, (2,{5},5) => 2, (2,{6},6) => 21, (3,{7},7) --------------------------------------------------------------------------------------------- => 14, (4,{9},9) => 1}, new BettiTally from {(0,{2},2) => 9, (1,{3},3) => 16, (1,{5},5) => 6, --------------------------------------------------------------------------------------------- (2,{4},4) => 3, (2,{5},5) => 3, (2,{6},6) => 16, (3,{7},7) => 9}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 9, (1,{3},3) => 15, (1,{4},4) => 3, (2,{4},4) => 3, (2,{6},6) => 15, (3,{7},7) --------------------------------------------------------------------------------------------- => 10, (3,{9},9) => 3, (4,{8},8) => 3, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 10, (1,{3},3) => 18, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 6, (2,{6},6) => 14, --------------------------------------------------------------------------------------------- (3,{7},7) => 7, (3,{9},9) => 2, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 8, --------------------------------------------------------------------------------------------- (1,{3},3) => 13, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 3, (2,{6},6) => 17, (3,{7},7) --------------------------------------------------------------------------------------------- => 12, (3,{9},9) => 1, (4,{8},8) => 3}, new BettiTally from {(0,{2},2) => 9, (1,{3},3) => 16, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) => 6, (2,{6},6) => 16, (3,{7},7) => 9}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 3, (1,{3},3) => 23, (2,{4},4) => 23, (2,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (1,{1},1) => 1, (1,{3},3) => 18, (2,{4},4) => 20, (2,{6},6) => 6, (3,{5},5) => 2, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 7, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 3, --------------------------------------------------------------------------------------------- (1,{1},1) => 2, (1,{3},3) => 21, (2,{4},4) => 21, (2,{6},6) => 3, (3,{5},5) => 1, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 4, (1,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 16, (2,{4},4) => 18, (2,{6},6) => 6, (3,{6},6) => 3, (3,{7},7) => 9, (4,{8},8) --------------------------------------------------------------------------------------------- => 3}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 6, (1,{1},1) => 3, (1,{3},3) => 24, --------------------------------------------------------------------------------------------- (2,{4},4) => 22, (3,{6},6) => 3, (3,{7},7) => 1}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 6, (1,{2},2) => 3, (1,{3},3) => 23, (2,{4},4) => 23, (3,{5},5) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 6, (1,{1},1) => 2, (1,{2},2) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 21, (2,{4},4) => 21, (3,{5},5) => 1, (3,{7},7) => 4, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{1},1) => 1, (1,{3},3) => 20, (2,{4},4) => 22, (2,{5},5) --------------------------------------------------------------------------------------------- => 2, (2,{6},6) => 3, (3,{5},5) => 6, (3,{7},7) => 5, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{1},1) => 3, (1,{3},3) => 22, (2,{4},4) => 24, (2,{6},6) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 6, (3,{7},7) => 3}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 3, (1,{3},3) => 23, --------------------------------------------------------------------------------------------- (2,{4},4) => 23, (2,{5},5) => 3, (3,{5},5) => 6, (3,{7},7) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{1},1) => 3, (1,{1},1) => 2, (1,{3},3) => 20, (2,{4},4) => 22, (2,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{5},5) => 4, (3,{7},7) => 5, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (0,{2},2) => 3, (1,{1},1) => 3, (1,{3},3) => 23, (2,{4},4) => 23, (3,{5},5) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (1,{1},1) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 18, (2,{4},4) => 20, (2,{6},6) => 3, (3,{5},5) => 2, (3,{7},7) => 7, (4,{8},8) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 3, (1,{1},1) => 5, --------------------------------------------------------------------------------------------- (1,{3},3) => 21, (2,{4},4) => 21, (3,{5},5) => 1, (3,{7},7) => 4, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (1,{1},1) => 6, (1,{3},3) => 16, (2,{4},4) --------------------------------------------------------------------------------------------- => 18, (2,{6},6) => 3, (3,{7},7) => 9, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 4, (1,{3},3) => 10, (1,{5},5) => 6, (2,{5},5) => 3, (2,{6},6) => 21, (3,{7},7) --------------------------------------------------------------------------------------------- => 15, (4,{8},8) => 2}, new BettiTally from {(0,{2},2) => 10, (1,{3},3) => 17, (1,{4},4) => --------------------------------------------------------------------------------------------- 3, (2,{4},4) => 4, (2,{6},6) => 14, (3,{7},7) => 8, (3,{9},9) => 3, (4,{8},8) => 2, --------------------------------------------------------------------------------------------- (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 9, (1,{3},3) => 15, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (1,{5},5) => 2, (2,{4},4) => 4, (2,{6},6) => 16, (3,{7},7) => 10, (3,{9},9) => 1, (4,{8},8) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 1, (1,{1},1) => 1, (1,{3},3) => 20, --------------------------------------------------------------------------------------------- (2,{4},4) => 21, (2,{6},6) => 5, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 5, (4,{8},8) => --------------------------------------------------------------------------------------------- 1}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 4, (1,{1},1) => 2, (1,{3},3) => 23, --------------------------------------------------------------------------------------------- (2,{4},4) => 22, (2,{6},6) => 2, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 4, (0,{2},2) => 1, (1,{1},1) => 3, (1,{3},3) => 18, (2,{4},4) --------------------------------------------------------------------------------------------- => 19, (2,{6},6) => 5, (3,{6},6) => 3, (3,{7},7) => 7, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{2},2) => 6, (1,{1},1) => 2, (1,{2},2) => 2, (1,{3},3) => 23, (2,{4},4) --------------------------------------------------------------------------------------------- => 22, (3,{5},5) => 1, (3,{6},6) => 1, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{1},1) => 1, (0,{2},2) => 1, (1,{3},3) => 22, (2,{4},4) => 23, (2,{5},5) => 2, (2,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{5},5) => 6, (3,{7},7) => 3}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{1},1) => 2, (1,{3},3) => 22, (2,{4},4) => 23, (2,{6},6) => 2, (3,{5},5) --------------------------------------------------------------------------------------------- => 4, (3,{7},7) => 3}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 1, --------------------------------------------------------------------------------------------- (1,{1},1) => 4, (1,{3},3) => 20, (2,{4},4) => 21, (2,{6},6) => 2, (3,{5},5) => 2, (3,{7},7) --------------------------------------------------------------------------------------------- => 5, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (0,{2},2) => 1, --------------------------------------------------------------------------------------------- (1,{1},1) => 6, (1,{3},3) => 18, (2,{4},4) => 19, (2,{6},6) => 2, (3,{7},7) => 7, (4,{8},8) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 2, (1,{3},3) => 7, (1,{5},5) => 6, (2,{5},5) => 2, --------------------------------------------------------------------------------------------- (2,{6},6) => 25, (3,{7},7) => 18, (4,{8},8) => 1, (4,{9},9) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 5, (1,{3},3) => 12, (1,{5},5) => 6, (2,{4},4) => 1, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 20, (3,{7},7) => 13, (4,{8},8) => 1}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 11, (1,{3},3) => 19, (1,{4},4) => 3, (2,{4},4) => 5, (2,{6},6) => 13, (3,{7},7) => 6, --------------------------------------------------------------------------------------------- (3,{9},9) => 3, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 10, --------------------------------------------------------------------------------------------- (1,{3},3) => 17, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 5, (2,{6},6) => 15, (3,{7},7) --------------------------------------------------------------------------------------------- => 8, (3,{9},9) => 1, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 2, --------------------------------------------------------------------------------------------- (1,{1},1) => 1, (1,{3},3) => 22, (2,{4},4) => 22, (2,{6},6) => 4, (3,{5},5) => 2, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 3}, new BettiTally from {(0,{0},0) => 4, (0,{2},2) => 2, (1,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 20, (2,{4},4) => 20, (2,{6},6) => 4, (3,{6},6) => 3, (3,{7},7) => 5, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 22, (2,{4},4) => 22, (2,{6},6) => 1, (3,{5},5) => 2, (3,{7},7) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 6, (1,{3},3) => --------------------------------------------------------------------------------------------- 20, (2,{4},4) => 20, (2,{6},6) => 1, (3,{7},7) => 5, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 1, (1,{3},3) => 9, (1,{5},5) => 6, (2,{4},4) => 1, (2,{5},5) => --------------------------------------------------------------------------------------------- 2, (2,{6},6) => 24, (3,{7},7) => 16, (4,{9},9) => 1}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 6, (1,{3},3) => 14, (1,{5},5) => 6, (2,{4},4) => 2, (2,{5},5) => 3, (2,{6},6) => --------------------------------------------------------------------------------------------- 19, (3,{7},7) => 11}, new BettiTally from {(0,{2},2) => 12, (1,{3},3) => 21, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (2,{4},4) => 6, (2,{6},6) => 12, (3,{7},7) => 4, (3,{9},9) => 3, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 10, (1,{3},3) => 16, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 15, (3,{7},7) => 9, (3,{9},9) => 2, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 11, (1,{3},3) => 19, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 6, (2,{6},6) --------------------------------------------------------------------------------------------- => 14, (3,{7},7) => 6, (3,{9},9) => 1}, new BettiTally from {(0,{0},0) => 3, (1,{3},3) => 21, --------------------------------------------------------------------------------------------- (2,{4},4) => 22, (2,{6},6) => 6, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 4, (1,{1},1) => 2, (1,{3},3) => 19, (2,{4},4) => 20, (2,{6},6) --------------------------------------------------------------------------------------------- => 6, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 6, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 4, (0,{2},2) => 3, (1,{1},1) => 3, (1,{3},3) => 22, (2,{4},4) => 21, (2,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{6},6) => 3, (3,{7},7) => 3}, new BettiTally from {(0,{0},0) => 4, (0,{2},2) => 6, --------------------------------------------------------------------------------------------- (1,{1},1) => 3, (1,{2},2) => 3, (1,{3},3) => 22, (2,{4},4) => 21, (3,{7},7) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 3, (1,{3},3) => 21, (2,{4},4) => 22, (2,{5},5) => 3, (2,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{5},5) => 6, (3,{7},7) => 4}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{1},1) => 3, (1,{3},3) => 21, (2,{4},4) => 22, (2,{6},6) => 3, (3,{5},5) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 4}, new BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (1,{1},1) => 5, (1,{3},3) => 19, --------------------------------------------------------------------------------------------- (2,{4},4) => 20, (2,{6},6) => 3, (3,{5},5) => 1, (3,{7},7) => 6, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (0,{2},2) => 3, (1,{1},1) => 6, (1,{3},3) => --------------------------------------------------------------------------------------------- 22, (2,{4},4) => 21, (3,{7},7) => 3}, new BettiTally from {(0,{2},2) => 11, (1,{3},3) => 18, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 4, (2,{6},6) => 14, (3,{7},7) => 7, (3,{9},9) => --------------------------------------------------------------------------------------------- 2, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 4, (0,{2},2) => 1, (1,{1},1) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 21, (2,{4},4) => 21, (2,{6},6) => 5, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 4}, new BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 5, --------------------------------------------------------------------------------------------- (1,{3},3) => 21, (2,{4},4) => 21, (2,{6},6) => 2, (3,{5},5) => 1, (3,{7},7) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 2, (1,{3},3) => 10, (1,{5},5) => 6, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 23, (3,{7},7) => 15, (4,{8},8) => 1}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 12, (1,{3},3) => 20, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 5, (2,{6},6) => 13, --------------------------------------------------------------------------------------------- (3,{7},7) => 5, (3,{9},9) => 2, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{3},3) => 12, (1,{5},5) => 6, (2,{4},4) => 1, (2,{5},5) => 3, (2,{6},6) => --------------------------------------------------------------------------------------------- 22, (3,{7},7) => 13}, new BettiTally from {(0,{2},2) => 13, (1,{3},3) => 22, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (1,{5},5) => 1, (2,{4},4) => 6, (2,{6},6) => 12, (3,{7},7) => 3, (3,{9},9) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 5, (1,{1},1) => 3, (1,{3},3) => 20, (2,{4},4) => 20, (2,{6},6) --------------------------------------------------------------------------------------------- => 6, (3,{6},6) => 3, (3,{7},7) => 5}, new BettiTally from {(0,{0},0) => 5, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{1},1) => 6, (1,{3},3) => 20, (2,{4},4) => 20, (2,{6},6) => 3, (3,{7},7) => 5}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 14, (1,{3},3) => 23, (1,{4},4) => 3, (2,{4},4) => 5, (2,{6},6) --------------------------------------------------------------------------------------------- => 11, (3,{7},7) => 2, (3,{9},9) => 3, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 10, (1,{5},5) => 6, (2,{5},5) => 3, (2,{6},6) => 25, (3,{7},7) => 15}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 15, (1,{3},3) => 25, (1,{4},4) => 3, (2,{4},4) => 6, (2,{6},6) --------------------------------------------------------------------------------------------- => 10, (3,{9},9) => 3}}; ---------------------- ---------------------- ---------------------- if g == 19 then return {new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 3, (1,{3},3) => 20, (2,{4},4) => 23, --------------------------------------------------------------------------------------------- (2,{6},6) => 4, (3,{5},5) => 4, (3,{6},6) => 3, (3,{7},7) => 8, (4,{8},8) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (1,{1},1) => 1, (1,{3},3) => 15, (2,{4},4) => 20, (2,{6},6) --------------------------------------------------------------------------------------------- => 7, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) => 13, (4,{8},8) => 6}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 6, (1,{1},1) => 1, (1,{3},3) => 23, (2,{4},4) => 24, (2,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) => 5, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 3, (1,{1},1) => 2, (1,{3},3) => 18, (2,{4},4) => 21, (2,{6},6) --------------------------------------------------------------------------------------------- => 4, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 10, (4,{8},8) => 5}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (1,{1},1) => 3, (1,{3},3) => 13, (2,{4},4) => 18, (2,{6},6) => 7, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 15, (4,{8},8) => 7}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 6, (1,{1},1) => 3, (1,{3},3) => 21, (2,{4},4) => 22, (2,{6},6) => 1, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 7, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{1},1) => 4, (1,{3},3) => 16, (2,{4},4) => 19, (2,{6},6) => 4, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 12, (4,{8},8) => 6}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 7, --------------------------------------------------------------------------------------------- (1,{1},1) => 1, (1,{2},2) => 1, (1,{3},3) => 23, (2,{4},4) => 24, (3,{5},5) => 3, (3,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{7},7) => 5, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 7, --------------------------------------------------------------------------------------------- (1,{1},1) => 3, (1,{2},2) => 1, (1,{3},3) => 21, (2,{4},4) => 22, (3,{5},5) => 1, (3,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{7},7) => 7, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 17, (2,{4},4) => 22, (2,{5},5) => 2, (2,{6},6) => 4, (3,{5},5) => 7, (3,{7},7) --------------------------------------------------------------------------------------------- => 11, (4,{8},8) => 5}, new BettiTally from {(0,{1},1) => 3, (1,{3},3) => 19, (2,{4},4) => --------------------------------------------------------------------------------------------- 24, (2,{6},6) => 4, (3,{5},5) => 7, (3,{7},7) => 9, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (1,{3},3) => 20, (2,{4},4) => 25, (2,{5},5) => 3, (2,{6},6) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 7, (3,{7},7) => 8, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{3},3) => 20, (2,{4},4) => 23, (2,{5},5) => 3, (2,{6},6) => 1, (3,{5},5) --------------------------------------------------------------------------------------------- => 7, (3,{7},7) => 8, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 22, (2,{4},4) => 27, (2,{5},5) => 1, (2,{6},6) => 3, (3,{5},5) => 7, (3,{7},7) --------------------------------------------------------------------------------------------- => 6, (4,{10},10) => 1}, new BettiTally from {(0,{1},1) => 2, (0,{2},2) => 3, (1,{3},3) => --------------------------------------------------------------------------------------------- 22, (2,{4},4) => 25, (2,{5},5) => 1, (2,{6},6) => 1, (3,{5},5) => 7, (3,{7},7) => 6, --------------------------------------------------------------------------------------------- (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 3, (1,{1},1) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 22, (2,{4},4) => 27, (2,{6},6) => 3, (3,{5},5) => 6, (3,{7},7) => 6, (4,{10},10) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, (1,{1},1) => 3, (1,{3},3) => 20, --------------------------------------------------------------------------------------------- (2,{4},4) => 25, (2,{6},6) => 3, (3,{5},5) => 4, (3,{7},7) => 8, (4,{8},8) => 1, (4,{10},10) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (1,{1},1) => 5, (1,{3},3) => 18, --------------------------------------------------------------------------------------------- (2,{4},4) => 23, (2,{6},6) => 3, (3,{5},5) => 2, (3,{7},7) => 10, (4,{8},8) => 2, (4,{10},10) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (1,{1},1) => 7, (1,{3},3) => 16, --------------------------------------------------------------------------------------------- (2,{4},4) => 21, (2,{6},6) => 3, (3,{7},7) => 12, (4,{8},8) => 3, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 1, (1,{3},3) => 7, (1,{5},5) => 5, (2,{4},4) => --------------------------------------------------------------------------------------------- 2, (2,{6},6) => 26, (3,{7},7) => 21, (3,{9},9) => 2, (4,{8},8) => 1, (4,{9},9) => 3, --------------------------------------------------------------------------------------------- (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 1, (1,{3},3) => 5, (1,{5},5) => 7, --------------------------------------------------------------------------------------------- (2,{4},4) => 2, (2,{6},6) => 28, (3,{7},7) => 23, (4,{8},8) => 1, (4,{9},9) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 4, (1,{3},3) => 7, (1,{5},5) => 7, (2,{5},5) => 2, (2,{6},6) => --------------------------------------------------------------------------------------------- 24, (3,{7},7) => 21, (4,{8},8) => 4, (4,{9},9) => 1}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{3},3) => 7, (1,{5},5) => 7, (2,{4},4) => 2, (2,{5},5) => 2, (2,{6},6) => --------------------------------------------------------------------------------------------- 26, (3,{7},7) => 21, (4,{8},8) => 1, (4,{9},9) => 1, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 5, (1,{3},3) => 10, (1,{5},5) => 7, (2,{4},4) => 3, (2,{5},5) => 1, (2,{6},6) --------------------------------------------------------------------------------------------- => 23, (3,{7},7) => 18, (4,{8},8) => 1, (4,{9},9) => 2}, new BettiTally from {(0,{2},2) => 6, --------------------------------------------------------------------------------------------- (1,{3},3) => 12, (1,{5},5) => 7, (2,{4},4) => 3, (2,{5},5) => 3, (2,{6},6) => 21, (3,{7},7) --------------------------------------------------------------------------------------------- => 16, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 8, (1,{3},3) => --------------------------------------------------------------------------------------------- 16, (1,{4},4) => 3, (2,{4},4) => 6, (2,{6},6) => 17, (3,{7},7) => 12, (3,{9},9) => 4, --------------------------------------------------------------------------------------------- (4,{8},8) => 1, (4,{10},10) => 3}, new BettiTally from {(0,{2},2) => 6, (1,{3},3) => 11, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 3, (2,{6},6) => 20, (3,{7},7) => 17, (3,{9},9) --------------------------------------------------------------------------------------------- => 3, (4,{8},8) => 4, (4,{10},10) => 2}, new BettiTally from {(0,{2},2) => 4, (1,{3},3) => 6, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 2, (2,{6},6) => 23, (3,{7},7) => 22, (3,{9},9) => 2, (4,{8},8) --------------------------------------------------------------------------------------------- => 7, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 7, (1,{3},3) => 14, (1,{4},4) => --------------------------------------------------------------------------------------------- 3, (1,{5},5) => 2, (2,{4},4) => 6, (2,{6},6) => 19, (3,{7},7) => 14, (3,{9},9) => 2, --------------------------------------------------------------------------------------------- (4,{8},8) => 1, (4,{10},10) => 2}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 9, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) => 3, (2,{6},6) => 22, (3,{7},7) => 19, (3,{9},9) --------------------------------------------------------------------------------------------- => 1, (4,{8},8) => 4, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 3, (1,{3},3) => 4, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 4, (2,{6},6) => 25, (3,{7},7) => 24, (4,{8},8) => 7}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 6, (1,{3},3) => 12, (1,{4},4) => 3, (1,{5},5) => 4, (2,{4},4) --------------------------------------------------------------------------------------------- => 6, (2,{6},6) => 21, (3,{7},7) => 16, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally --------------------------------------------------------------------------------------------- from {(0,{0},0) => 1, (0,{2},2) => 4, (1,{3},3) => 22, (2,{4},4) => 24, (2,{6},6) => 3, --------------------------------------------------------------------------------------------- (3,{5},5) => 4, (3,{6},6) => 3, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 1, (1,{1},1) => 1, (1,{3},3) => 17, (2,{4},4) => 21, (2,{6},6) --------------------------------------------------------------------------------------------- => 6, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) => 11, (4,{8},8) => 5}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 7, (1,{1},1) => 1, (1,{3},3) => 25, (2,{4},4) => 25, (3,{5},5) --------------------------------------------------------------------------------------------- => 3, (3,{6},6) => 3, (3,{7},7) => 3, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 4, (1,{1},1) => 2, (1,{3},3) => 20, (2,{4},4) => 22, (2,{6},6) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{6},6) => 3, (3,{7},7) => 8, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{1},1) => 3, (1,{3},3) => 15, (2,{4},4) => 19, (2,{6},6) => 6, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 13, (4,{8},8) => 6}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 7, (1,{1},1) => 3, (1,{3},3) => 23, (2,{4},4) => 23, (3,{5},5) => 1, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 5, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 4, --------------------------------------------------------------------------------------------- (1,{1},1) => 4, (1,{3},3) => 18, (2,{4},4) => 20, (2,{6},6) => 3, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 10, (4,{8},8) => 5}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 7, (1,{2},2) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 22, (2,{4},4) => 24, (3,{5},5) => 4, (3,{7},7) => 6, (4,{8},8) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 7, (1,{1},1) => 2, (1,{2},2) => 3, (1,{3},3) => --------------------------------------------------------------------------------------------- 20, (2,{4},4) => 22, (3,{5},5) => 2, (3,{7},7) => 8, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{1},1) => 1, (0,{2},2) => 1, (1,{3},3) => 19, (2,{4},4) => 23, (2,{5},5) --------------------------------------------------------------------------------------------- => 2, (2,{6},6) => 3, (3,{5},5) => 7, (3,{7},7) => 9, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{1},1) => 3, (0,{2},2) => 1, (1,{3},3) => 21, (2,{4},4) => 25, (2,{6},6) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 7, (3,{7},7) => 7, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 22, (2,{4},4) => 26, (2,{5},5) => 3, (2,{6},6) => 2, (3,{5},5) => 7, (3,{7},7) --------------------------------------------------------------------------------------------- => 6, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 4, (1,{3},3) => --------------------------------------------------------------------------------------------- 22, (2,{4},4) => 24, (2,{5},5) => 3, (3,{5},5) => 7, (3,{7},7) => 6, (4,{8},8) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{1},1) => 2, (0,{2},2) => 4, (1,{3},3) => 24, (2,{4},4) => 26, (2,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{5},5) => 7, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 3, (1,{3},3) => 22, (2,{4},4) => 26, (2,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{5},5) => 4, (3,{7},7) => 6, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 5, (1,{3},3) => 20, (2,{4},4) => 24, (2,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{5},5) => 2, (3,{7},7) => 8, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 4, (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 7, (1,{3},3) => 18, (2,{4},4) --------------------------------------------------------------------------------------------- => 22, (2,{6},6) => 2, (3,{7},7) => 10, (4,{8},8) => 2, (4,{10},10) => 1}, new BettiTally --------------------------------------------------------------------------------------------- from {(0,{0},0) => 2, (1,{3},3) => 10, (2,{4},4) => 2, (2,{6},6) => 24, (3,{7},7) => 18, --------------------------------------------------------------------------------------------- (3,{9},9) => 7, (4,{9},9) => 3, (4,{10},10) => 4}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{3},3) => 11, (1,{5},5) => 3, (2,{4},4) => 3, (2,{6},6) => 23, (3,{7},7) --------------------------------------------------------------------------------------------- => 17, (3,{9},9) => 4, (4,{9},9) => 3, (4,{10},10) => 2}, new BettiTally from {(0,{0},0) => --------------------------------------------------------------------------------------------- 1, (0,{2},2) => 2, (1,{3},3) => 9, (1,{5},5) => 5, (2,{4},4) => 3, (2,{6},6) => 25, (3,{7},7) --------------------------------------------------------------------------------------------- => 19, (3,{9},9) => 2, (4,{9},9) => 3, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => --------------------------------------------------------------------------------------------- 1, (1,{3},3) => 4, (1,{5},5) => 7, (2,{5},5) => 1, (2,{6},6) => 28, (3,{7},7) => 24, --------------------------------------------------------------------------------------------- (4,{8},8) => 3, (4,{9},9) => 2}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 6, (1,{5},5) => 7, (2,{5},5) => 3, (2,{6},6) => 26, (3,{7},7) => 22, (4,{8},8) --------------------------------------------------------------------------------------------- => 3, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 1, (1,{3},3) => 7, --------------------------------------------------------------------------------------------- (1,{5},5) => 7, (2,{4},4) => 3, (2,{6},6) => 27, (3,{7},7) => 21, (4,{9},9) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 5, (1,{3},3) => 9, (1,{5},5) => 7, (2,{4},4) => 1, (2,{5},5) => --------------------------------------------------------------------------------------------- 2, (2,{6},6) => 23, (3,{7},7) => 19, (4,{8},8) => 3, (4,{9},9) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 9, (1,{5},5) => 7, (2,{4},4) => 3, (2,{5},5) => --------------------------------------------------------------------------------------------- 2, (2,{6},6) => 25, (3,{7},7) => 19, (4,{9},9) => 1, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 6, (1,{3},3) => 12, (1,{5},5) => 7, (2,{4},4) => 4, (2,{5},5) => 1, (2,{6},6) --------------------------------------------------------------------------------------------- => 22, (3,{7},7) => 16, (4,{9},9) => 2}, new BettiTally from {(0,{2},2) => 7, (1,{3},3) => --------------------------------------------------------------------------------------------- 14, (1,{5},5) => 7, (2,{4},4) => 4, (2,{5},5) => 3, (2,{6},6) => 20, (3,{7},7) => 14, --------------------------------------------------------------------------------------------- (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 6, (1,{3},3) => 10, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (2,{4},4) => 1, (2,{6},6) => 20, (3,{7},7) => 18, (3,{9},9) => 4, (4,{8},8) => 6, (4,{10},10) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{2},2) => 9, (1,{3},3) => 18, (1,{4},4) => 3, (2,{4},4) => 7, --------------------------------------------------------------------------------------------- (2,{6},6) => 16, (3,{7},7) => 10, (3,{9},9) => 4, (4,{10},10) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 7, (1,{3},3) => 13, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 4, (2,{6},6) --------------------------------------------------------------------------------------------- => 19, (3,{7},7) => 15, (3,{9},9) => 3, (4,{8},8) => 3, (4,{10},10) => 2}, new BettiTally --------------------------------------------------------------------------------------------- from {(0,{2},2) => 5, (1,{3},3) => 8, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 1, --------------------------------------------------------------------------------------------- (2,{6},6) => 22, (3,{7},7) => 20, (3,{9},9) => 2, (4,{8},8) => 6, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 8, (1,{3},3) => 16, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) --------------------------------------------------------------------------------------------- => 7, (2,{6},6) => 18, (3,{7},7) => 12, (3,{9},9) => 2, (4,{10},10) => 2}, new BettiTally --------------------------------------------------------------------------------------------- from {(0,{2},2) => 6, (1,{3},3) => 11, (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) => 4, --------------------------------------------------------------------------------------------- (2,{6},6) => 21, (3,{7},7) => 17, (3,{9},9) => 1, (4,{8},8) => 3, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 4, (1,{3},3) => 6, (1,{4},4) => 2, (1,{5},5) => 4, (2,{6},6) => --------------------------------------------------------------------------------------------- 24, (3,{7},7) => 22, (3,{8},8) => 1, (4,{8},8) => 7}, new BettiTally from {(0,{2},2) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 6, (1,{4},4) => 3, (1,{5},5) => 4, (2,{4},4) => 1, (2,{6},6) => 24, (3,{7},7) => --------------------------------------------------------------------------------------------- 22, (4,{8},8) => 6}, new BettiTally from {(0,{2},2) => 7, (1,{3},3) => 14, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (1,{5},5) => 4, (2,{4},4) => 7, (2,{6},6) => 20, (3,{7},7) => 14, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 5, (1,{3},3) => 24, (2,{4},4) => 25, (2,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{5},5) => 4, (3,{6},6) => 3, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 2, (1,{1},1) => 1, (1,{3},3) => 19, (2,{4},4) => 22, (2,{6},6) --------------------------------------------------------------------------------------------- => 5, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) => 9, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 5, (1,{1},1) => 2, (1,{3},3) => 22, (2,{4},4) => 23, (2,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{2},2) => 2, (1,{1},1) => 3, (1,{3},3) => 17, (2,{4},4) => 20, (2,{6},6) --------------------------------------------------------------------------------------------- => 5, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 11, (4,{8},8) => 5}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{2},2) => 5, (1,{1},1) => 4, (1,{3},3) => 20, (2,{4},4) => 21, (2,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{6},6) => 3, (3,{7},7) => 8, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 7, (1,{2},2) => 2, (1,{3},3) => 24, (2,{4},4) => 25, (3,{5},5) => 4, (3,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 7, --------------------------------------------------------------------------------------------- (1,{1},1) => 2, (1,{2},2) => 2, (1,{3},3) => 22, (2,{4},4) => 23, (3,{5},5) => 2, (3,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 7, --------------------------------------------------------------------------------------------- (1,{1},1) => 4, (1,{2},2) => 2, (1,{3},3) => 20, (2,{4},4) => 21, (3,{6},6) => 1, (3,{7},7) --------------------------------------------------------------------------------------------- => 8, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 1, (0,{2},2) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 21, (2,{4},4) => 24, (2,{5},5) => 2, (2,{6},6) => 2, (3,{5},5) => 7, (3,{7},7) --------------------------------------------------------------------------------------------- => 7, (4,{8},8) => 3}, new BettiTally from {(0,{1},1) => 3, (0,{2},2) => 2, (1,{3},3) => 23, --------------------------------------------------------------------------------------------- (2,{4},4) => 26, (2,{6},6) => 2, (3,{5},5) => 7, (3,{7},7) => 5, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 5, (1,{3},3) => --------------------------------------------------------------------------------------------- 22, (2,{4},4) => 25, (2,{6},6) => 1, (3,{5},5) => 2, (3,{7},7) => 6, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 1, (1,{3},3) => 6, (1,{5},5) => 7, (2,{4},4) => --------------------------------------------------------------------------------------------- 1, (2,{5},5) => 1, (2,{6},6) => 27, (3,{7},7) => 22, (4,{8},8) => 2, (4,{9},9) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 8, (1,{5},5) => 7, (2,{4},4) => --------------------------------------------------------------------------------------------- 1, (2,{5},5) => 3, (2,{6},6) => 25, (3,{7},7) => 20, (4,{8},8) => 2, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 6, (1,{3},3) => 11, (1,{5},5) => 7, (2,{4},4) => 2, (2,{5},5) --------------------------------------------------------------------------------------------- => 2, (2,{6},6) => 22, (3,{7},7) => 17, (4,{8},8) => 2, (4,{9},9) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 7, (1,{3},3) => 12, (1,{4},4) => 3, (2,{4},4) => 2, (2,{6},6) => 19, (3,{7},7) --------------------------------------------------------------------------------------------- => 16, (3,{9},9) => 4, (4,{8},8) => 5, (4,{10},10) => 2}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 8, (1,{3},3) => 15, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 5, (2,{6},6) => 18, --------------------------------------------------------------------------------------------- (3,{7},7) => 13, (3,{9},9) => 3, (4,{8},8) => 2, (4,{10},10) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 6, (1,{3},3) => 10, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 2, (2,{6},6) --------------------------------------------------------------------------------------------- => 21, (3,{7},7) => 18, (3,{9},9) => 2, (4,{8},8) => 5, (4,{10},10) => 1}, new BettiTally --------------------------------------------------------------------------------------------- from {(0,{2},2) => 7, (1,{3},3) => 13, (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) => 5, --------------------------------------------------------------------------------------------- (2,{6},6) => 20, (3,{7},7) => 15, (3,{9},9) => 1, (4,{8},8) => 2, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 5, (1,{3},3) => 8, (1,{4},4) => 3, (1,{5},5) => 4, (2,{4},4) => --------------------------------------------------------------------------------------------- 2, (2,{6},6) => 23, (3,{7},7) => 20, (4,{8},8) => 5}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 18, (2,{4},4) => 22, (2,{6},6) => 7, (3,{5},5) => 4, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 10, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 6, (1,{3},3) => 26, --------------------------------------------------------------------------------------------- (2,{4},4) => 26, (2,{6},6) => 1, (3,{5},5) => 4, (3,{6},6) => 3, (3,{7},7) => 2, (4,{8},8) => --------------------------------------------------------------------------------------------- 1}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 3, (1,{1},1) => 1, (1,{3},3) => 21, --------------------------------------------------------------------------------------------- (2,{4},4) => 23, (2,{6},6) => 4, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) => 7, (4,{8},8) => --------------------------------------------------------------------------------------------- 3}, new BettiTally from {(0,{0},0) => 3, (1,{1},1) => 2, (1,{3},3) => 16, (2,{4},4) => 20, --------------------------------------------------------------------------------------------- (2,{6},6) => 7, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 12, (4,{8},8) => 5}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 6, (1,{1},1) => 2, (1,{3},3) => 24, (2,{4},4) --------------------------------------------------------------------------------------------- => 24, (2,{6},6) => 1, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 4, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 3, (0,{2},2) => 3, (1,{1},1) => 3, (1,{3},3) => 19, (2,{4},4) --------------------------------------------------------------------------------------------- => 21, (2,{6},6) => 4, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 9, (4,{8},8) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 4, (1,{1},1) => 4, (1,{3},3) => 14, (2,{4},4) => 18, (2,{6},6) --------------------------------------------------------------------------------------------- => 7, (3,{6},6) => 3, (3,{7},7) => 14, (4,{8},8) => 6}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 6, (1,{1},1) => 4, (1,{3},3) => 22, (2,{4},4) => 22, (2,{6},6) => 1, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 7, --------------------------------------------------------------------------------------------- (1,{2},2) => 1, (1,{3},3) => 26, (2,{4},4) => 26, (3,{5},5) => 4, (3,{6},6) => 2, (3,{7},7) --------------------------------------------------------------------------------------------- => 2, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 7, (1,{1},1) => 2, --------------------------------------------------------------------------------------------- (1,{2},2) => 1, (1,{3},3) => 24, (2,{4},4) => 24, (3,{5},5) => 2, (3,{6},6) => 2, (3,{7},7) --------------------------------------------------------------------------------------------- => 4, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 7, (1,{1},1) => 4, --------------------------------------------------------------------------------------------- (1,{2},2) => 1, (1,{3},3) => 22, (2,{4},4) => 22, (3,{6},6) => 2, (3,{7},7) => 6, (4,{8},8) --------------------------------------------------------------------------------------------- => 3}, new BettiTally from {(0,{0},0) => 2, (1,{3},3) => 18, (2,{4},4) => 22, (2,{5},5) => 3, --------------------------------------------------------------------------------------------- (2,{6},6) => 4, (3,{5},5) => 7, (3,{7},7) => 10, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{1},1) => 2, (1,{3},3) => 20, (2,{4},4) => 24, (2,{5},5) => 1, (2,{6},6) --------------------------------------------------------------------------------------------- => 4, (3,{5},5) => 7, (3,{7},7) => 8, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{1},1) => 1, (0,{2},2) => 3, (1,{3},3) => 23, (2,{4},4) => 25, (2,{5},5) => 2, (2,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{5},5) => 7, (3,{7},7) => 5, (4,{8},8) => 2}, new BettiTally from {(0,{1},1) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{3},3) => 25, (2,{4},4) => 27, (2,{6},6) => 1, (3,{5},5) => 7, (3,{7},7) --------------------------------------------------------------------------------------------- => 3, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 3, (1,{1},1) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 20, (2,{4},4) => 24, (2,{6},6) => 4, (3,{5},5) => 6, (3,{7},7) => 8, (4,{8},8) --------------------------------------------------------------------------------------------- => 3}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, (1,{1},1) => 3, (1,{3},3) => 18, --------------------------------------------------------------------------------------------- (2,{4},4) => 22, (2,{6},6) => 4, (3,{5},5) => 4, (3,{7},7) => 10, (4,{8},8) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (1,{1},1) => 4, (1,{3},3) => 21, (2,{4},4) --------------------------------------------------------------------------------------------- => 25, (2,{6},6) => 3, (3,{5},5) => 3, (3,{7},7) => 7, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{1},1) => 3, (1,{1},1) => 5, (1,{3},3) => 16, (2,{4},4) => 20, (2,{6},6) --------------------------------------------------------------------------------------------- => 4, (3,{5},5) => 2, (3,{7},7) => 12, (4,{8},8) => 5}, new BettiTally from {(0,{0},0) => 4, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (1,{1},1) => 6, (1,{3},3) => 19, (2,{4},4) => 23, (2,{6},6) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{7},7) => 9, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 4, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (1,{1},1) => 7, (1,{3},3) => 14, (2,{4},4) => 18, (2,{6},6) => 4, (3,{7},7) --------------------------------------------------------------------------------------------- => 14, (4,{8},8) => 6}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 8, --------------------------------------------------------------------------------------------- (1,{5},5) => 6, (2,{4},4) => 2, (2,{6},6) => 26, (3,{7},7) => 20, (3,{9},9) => 1, (4,{8},8) --------------------------------------------------------------------------------------------- => 1, (4,{9},9) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 8, --------------------------------------------------------------------------------------------- (1,{5},5) => 7, (2,{4},4) => 2, (2,{5},5) => 1, (2,{6},6) => 26, (3,{7},7) => 20, (4,{8},8) --------------------------------------------------------------------------------------------- => 1, (4,{9},9) => 2}, new BettiTally from {(0,{2},2) => 6, (1,{3},3) => 10, (1,{5},5) => 7, --------------------------------------------------------------------------------------------- (2,{5},5) => 3, (2,{6},6) => 22, (3,{7},7) => 18, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 3, (1,{3},3) => 10, (1,{5},5) => 7, (2,{4},4) => 2, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 24, (3,{7},7) => 18, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally --------------------------------------------------------------------------------------------- from {(0,{2},2) => 7, (1,{3},3) => 13, (1,{5},5) => 7, (2,{4},4) => 3, (2,{5},5) => 2, --------------------------------------------------------------------------------------------- (2,{6},6) => 21, (3,{7},7) => 15, (4,{8},8) => 1, (4,{9},9) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 8, (1,{3},3) => 14, (1,{4},4) => 3, (2,{4},4) => 3, (2,{6},6) => 18, (3,{7},7) --------------------------------------------------------------------------------------------- => 14, (3,{9},9) => 4, (4,{8},8) => 4, (4,{10},10) => 2}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 9, (1,{3},3) => 17, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 6, (2,{6},6) => 17, --------------------------------------------------------------------------------------------- (3,{7},7) => 11, (3,{9},9) => 3, (4,{8},8) => 1, (4,{10},10) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 7, (1,{3},3) => 12, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 3, (2,{6},6) --------------------------------------------------------------------------------------------- => 20, (3,{7},7) => 16, (3,{9},9) => 2, (4,{8},8) => 4, (4,{10},10) => 1}, new BettiTally --------------------------------------------------------------------------------------------- from {(0,{2},2) => 5, (1,{3},3) => 7, (1,{4},4) => 3, (1,{5},5) => 3, (2,{6},6) => 23, --------------------------------------------------------------------------------------------- (3,{7},7) => 21, (3,{9},9) => 1, (4,{8},8) => 7}, new BettiTally from {(0,{2},2) => 8, --------------------------------------------------------------------------------------------- (1,{3},3) => 15, (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) => 6, (2,{6},6) => 19, (3,{7},7) --------------------------------------------------------------------------------------------- => 13, (3,{9},9) => 1, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 6, (1,{3},3) => 10, (1,{4},4) => 3, (1,{5},5) => 4, (2,{4},4) => 3, (2,{6},6) => 22, --------------------------------------------------------------------------------------------- (3,{7},7) => 18, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 20, (2,{4},4) => 23, (2,{6},6) => 6, (3,{5},5) => 4, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 8, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 7, (1,{3},3) => 28, --------------------------------------------------------------------------------------------- (2,{4},4) => 27, (3,{5},5) => 4, (3,{6},6) => 3}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 4, (1,{1},1) => 1, (1,{3},3) => 23, (2,{4},4) => 24, (2,{6},6) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 3, (3,{6},6) => 3, (3,{7},7) => 5, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{1},1) => 2, (1,{3},3) => 18, (2,{4},4) => 21, (2,{6},6) => 6, (3,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{6},6) => 3, (3,{7},7) => 10, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 7, (1,{1},1) => 2, (1,{3},3) => 26, (2,{4},4) => 25, (3,{5},5) => 2, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 2, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 4, --------------------------------------------------------------------------------------------- (1,{1},1) => 3, (1,{3},3) => 21, (2,{4},4) => 22, (2,{6},6) => 3, (3,{5},5) => 1, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 7, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 4, (0,{2},2) => 1, --------------------------------------------------------------------------------------------- (1,{1},1) => 4, (1,{3},3) => 16, (2,{4},4) => 19, (2,{6},6) => 6, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 12, (4,{8},8) => 5}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 7, (1,{1},1) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 24, (2,{4},4) => 23, (3,{6},6) => 3, (3,{7},7) => 4, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 7, (1,{1},1) => 1, (1,{2},2) => 3, (1,{3},3) => --------------------------------------------------------------------------------------------- 23, (2,{4},4) => 24, (3,{5},5) => 3, (3,{7},7) => 5, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{2},2) => 7, (1,{1},1) => 3, (1,{2},2) => 3, (1,{3},3) => 21, (2,{4},4) --------------------------------------------------------------------------------------------- => 22, (3,{5},5) => 1, (3,{7},7) => 7, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{3},3) => 20, (2,{4},4) => 23, (2,{5},5) => 3, (2,{6},6) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 7, (3,{7},7) => 8, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{3},3) => 22, (2,{4},4) => 25, (2,{5},5) => 1, (2,{6},6) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 7, (3,{7},7) => 6, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 4, (1,{3},3) => 25, (2,{4},4) => 26, (2,{5},5) => 2, (3,{5},5) => 7, (3,{7},7) --------------------------------------------------------------------------------------------- => 3, (4,{8},8) => 1}, new BettiTally from {(0,{1},1) => 3, (0,{2},2) => 4, (1,{3},3) => 27, --------------------------------------------------------------------------------------------- (2,{4},4) => 28, (3,{5},5) => 7, (3,{7},7) => 1}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 1, (1,{3},3) => 22, (2,{4},4) => 25, (2,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{5},5) => 6, (3,{7},7) => 6, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 3, (1,{3},3) => 20, (2,{4},4) => 23, (2,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{5},5) => 4, (3,{7},7) => 8, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 5, (1,{3},3) => 18, (2,{4},4) => 21, (2,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{5},5) => 2, (3,{7},7) => 10, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 4, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 6, (1,{3},3) => 21, (2,{4},4) => 24, (2,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{5},5) => 1, (3,{7},7) => 7, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 4, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 7, (1,{3},3) => 16, (2,{4},4) => 19, (2,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 12, (4,{8},8) => 5}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 11, (1,{5},5) => 1, (2,{4},4) => 2, (2,{6},6) => 24, (3,{7},7) => 17, (3,{9},9) --------------------------------------------------------------------------------------------- => 6, (4,{9},9) => 3, (4,{10},10) => 3}, new BettiTally from {(0,{0},0) => 2, (1,{3},3) => 9, --------------------------------------------------------------------------------------------- (1,{5},5) => 3, (2,{4},4) => 2, (2,{6},6) => 26, (3,{7},7) => 19, (3,{9},9) => 4, (4,{9},9) --------------------------------------------------------------------------------------------- => 3, (4,{10},10) => 2}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 3, (1,{3},3) => --------------------------------------------------------------------------------------------- 10, (1,{5},5) => 6, (2,{4},4) => 3, (2,{6},6) => 25, (3,{7},7) => 18, (3,{9},9) => 1, --------------------------------------------------------------------------------------------- (4,{9},9) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 7, --------------------------------------------------------------------------------------------- (1,{5},5) => 7, (2,{5},5) => 2, (2,{6},6) => 26, (3,{7},7) => 21, (4,{8},8) => 3, (4,{9},9) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 3, (1,{3},3) => 10, (1,{5},5) => 7, --------------------------------------------------------------------------------------------- (2,{4},4) => 3, (2,{5},5) => 1, (2,{6},6) => 25, (3,{7},7) => 18, (4,{9},9) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 7, (1,{3},3) => 12, (1,{5},5) => 7, (2,{4},4) => 1, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 21, (3,{7},7) => 16, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 4, (1,{3},3) => 12, (1,{5},5) => 7, (2,{4},4) => 3, (2,{5},5) => 3, (2,{6},6) => --------------------------------------------------------------------------------------------- 23, (3,{7},7) => 16, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 8, (1,{3},3) => 15, --------------------------------------------------------------------------------------------- (1,{5},5) => 7, (2,{4},4) => 4, (2,{5},5) => 2, (2,{6},6) => 20, (3,{7},7) => 13, (4,{9},9) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{2},2) => 9, (1,{3},3) => 16, (1,{4},4) => 3, (2,{4},4) => 4, --------------------------------------------------------------------------------------------- (2,{6},6) => 17, (3,{7},7) => 12, (3,{9},9) => 4, (4,{8},8) => 3, (4,{10},10) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 10, (1,{3},3) => 19, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) --------------------------------------------------------------------------------------------- => 7, (2,{6},6) => 16, (3,{7},7) => 9, (3,{9},9) => 3, (4,{10},10) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 8, (1,{3},3) => 14, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 4, (2,{6},6) --------------------------------------------------------------------------------------------- => 19, (3,{7},7) => 14, (3,{9},9) => 2, (4,{8},8) => 3, (4,{10},10) => 1}, new BettiTally --------------------------------------------------------------------------------------------- from {(0,{2},2) => 6, (1,{3},3) => 9, (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) => 1, --------------------------------------------------------------------------------------------- (2,{6},6) => 22, (3,{7},7) => 19, (3,{9},9) => 1, (4,{8},8) => 6}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 9, (1,{3},3) => 17, (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) => 7, (2,{6},6) --------------------------------------------------------------------------------------------- => 18, (3,{7},7) => 11, (3,{9},9) => 1, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 7, (1,{3},3) => 12, (1,{4},4) => 3, (1,{5},5) => 4, (2,{4},4) => 4, (2,{6},6) => 21, --------------------------------------------------------------------------------------------- (3,{7},7) => 16, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 22, (2,{4},4) => 24, (2,{6},6) => 5, (3,{5},5) => 4, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 6, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 5, (1,{1},1) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 25, (2,{4},4) => 25, (2,{6},6) => 2, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 3, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 2, (1,{1},1) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 20, (2,{4},4) => 22, (2,{6},6) => 5, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 8, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 5, (1,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 23, (2,{4},4) => 23, (2,{6},6) => 2, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 5, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 4, (0,{2},2) => 2, (1,{1},1) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 18, (2,{4},4) => 20, (2,{6},6) => 5, (3,{6},6) => 3, (3,{7},7) => 10, (4,{8},8) --------------------------------------------------------------------------------------------- => 4}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 7, (1,{1},1) => 1, (1,{2},2) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 25, (2,{4},4) => 25, (3,{5},5) => 3, (3,{6},6) => 1, (3,{7},7) => 3, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 7, (1,{1},1) => 3, (1,{2},2) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 23, (2,{4},4) => 23, (3,{5},5) => 1, (3,{6},6) => 1, (3,{7},7) => 5, (4,{8},8) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{1},1) => 2, (0,{2},2) => 2, (1,{3},3) => 20, (2,{4},4) => 24, --------------------------------------------------------------------------------------------- (2,{5},5) => 1, (2,{6},6) => 2, (3,{5},5) => 7, (3,{7},7) => 8, (4,{8},8) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 2, (1,{3},3) => 22, (2,{4},4) => 24, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 2, (3,{5},5) => 7, (3,{7},7) => 6, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{1},1) => 2, (0,{2},2) => 2, (1,{3},3) => 24, (2,{4},4) => 26, (2,{5},5) --------------------------------------------------------------------------------------------- => 1, (2,{6},6) => 2, (3,{5},5) => 7, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 1, (1,{3},3) => 24, (2,{4},4) --------------------------------------------------------------------------------------------- => 26, (2,{6},6) => 2, (3,{5},5) => 6, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 3, (1,{3},3) => 22, (2,{4},4) --------------------------------------------------------------------------------------------- => 24, (2,{6},6) => 2, (3,{5},5) => 4, (3,{7},7) => 6, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 5, (1,{3},3) => 20, (2,{4},4) --------------------------------------------------------------------------------------------- => 22, (2,{6},6) => 2, (3,{5},5) => 2, (3,{7},7) => 8, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 4, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 7, (1,{3},3) => 18, (2,{4},4) --------------------------------------------------------------------------------------------- => 20, (2,{6},6) => 2, (3,{7},7) => 10, (4,{8},8) => 4}, new BettiTally from {(0,{2},2) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 8, (1,{5},5) => 7, (2,{4},4) => 2, (2,{5},5) => 1, (2,{6},6) => 24, (3,{7},7) => --------------------------------------------------------------------------------------------- 20, (4,{8},8) => 2, (4,{9},9) => 2}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 9, (1,{5},5) => 7, (2,{4},4) => 1, (2,{5},5) => 2, (2,{6},6) => 25, (3,{7},7) => --------------------------------------------------------------------------------------------- 19, (4,{8},8) => 2, (4,{9},9) => 1}, new BettiTally from {(0,{2},2) => 8, (1,{3},3) => 14, --------------------------------------------------------------------------------------------- (1,{5},5) => 7, (2,{4},4) => 2, (2,{5},5) => 3, (2,{6},6) => 20, (3,{7},7) => 14, (4,{8},8) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{2},2) => 10, (1,{3},3) => 18, (1,{4},4) => 3, (2,{4},4) => 5, --------------------------------------------------------------------------------------------- (2,{6},6) => 16, (3,{7},7) => 10, (3,{9},9) => 4, (4,{8},8) => 2, (4,{10},10) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 8, (1,{3},3) => 13, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) --------------------------------------------------------------------------------------------- => 2, (2,{6},6) => 19, (3,{7},7) => 15, (3,{9},9) => 3, (4,{8},8) => 5, (4,{10},10) => 1}, --------------------------------------------------------------------------------------------- new BettiTally from {(0,{2},2) => 9, (1,{3},3) => 16, (1,{4},4) => 3, (1,{5},5) => 2, --------------------------------------------------------------------------------------------- (2,{4},4) => 5, (2,{6},6) => 18, (3,{7},7) => 12, (3,{9},9) => 2, (4,{8},8) => 2, (4,{10},10) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{2},2) => 7, (1,{3},3) => 11, (1,{4},4) => 3, (1,{5},5) => 3, --------------------------------------------------------------------------------------------- (2,{4},4) => 2, (2,{6},6) => 21, (3,{7},7) => 17, (3,{9},9) => 1, (4,{8},8) => 5}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 8, (1,{3},3) => 14, (1,{4},4) => 3, (1,{5},5) => 4, (2,{4},4) --------------------------------------------------------------------------------------------- => 5, (2,{6},6) => 20, (3,{7},7) => 14, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{3},3) => 24, (2,{4},4) => 25, (2,{6},6) => 4, (3,{5},5) => 4, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, (1,{1},1) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 19, (2,{4},4) => 22, (2,{6},6) => 7, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 9, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 5, (1,{1},1) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 21, (2,{4},4) => 23, (2,{6},6) => 2, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 7, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 6, (1,{1},1) => 1, --------------------------------------------------------------------------------------------- (1,{3},3) => 27, (2,{4},4) => 26, (2,{6},6) => 1, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 3, (1,{1},1) => 2, (1,{3},3) => 22, --------------------------------------------------------------------------------------------- (2,{4},4) => 23, (2,{6},6) => 4, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 6, (4,{8},8) => --------------------------------------------------------------------------------------------- 2}, new BettiTally from {(0,{0},0) => 4, (1,{1},1) => 3, (1,{3},3) => 17, (2,{4},4) => 20, --------------------------------------------------------------------------------------------- (2,{6},6) => 7, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 11, (4,{8},8) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 3, (0,{2},2) => 6, (1,{1},1) => 3, (1,{3},3) => 25, (2,{4},4) --------------------------------------------------------------------------------------------- => 24, (2,{6},6) => 1, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) => 3, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 4, (0,{2},2) => 3, (1,{1},1) => 4, (1,{3},3) => 20, (2,{4},4) --------------------------------------------------------------------------------------------- => 21, (2,{6},6) => 4, (3,{6},6) => 3, (3,{7},7) => 8, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 7, (1,{1},1) => 1, (1,{2},2) => 1, (1,{3},3) => 27, (2,{4},4) --------------------------------------------------------------------------------------------- => 26, (3,{5},5) => 3, (3,{6},6) => 2, (3,{7},7) => 1}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 7, (1,{1},1) => 3, (1,{2},2) => 1, (1,{3},3) => 25, (2,{4},4) => 24, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 2, (3,{7},7) => 3, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 7, (1,{1},1) => 1, (1,{2},2) => 2, (1,{3},3) => 21, (2,{4},4) => 23, (3,{5},5) --------------------------------------------------------------------------------------------- => 3, (3,{6},6) => 1, (3,{7},7) => 7, (4,{8},8) => 4}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{1},1) => 1, (1,{3},3) => 21, (2,{4},4) => 24, (2,{5},5) => 2, (2,{6},6) => 4, (3,{5},5) --------------------------------------------------------------------------------------------- => 7, (3,{7},7) => 7, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 23, (2,{4},4) => 26, (2,{6},6) => 4, (3,{5},5) => 7, (3,{7},7) => 5, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 3, (1,{3},3) => 24, (2,{4},4) => 25, --------------------------------------------------------------------------------------------- (2,{5},5) => 3, (2,{6},6) => 1, (3,{5},5) => 7, (3,{7},7) => 4, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{1},1) => 2, (0,{2},2) => 3, (1,{3},3) => 26, (2,{4},4) --------------------------------------------------------------------------------------------- => 27, (2,{5},5) => 1, (2,{6},6) => 1, (3,{5},5) => 7, (3,{7},7) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{1},1) => 3, (0,{2},2) => 3, (1,{1},1) => 1, (1,{3},3) => 26, (2,{4},4) --------------------------------------------------------------------------------------------- => 27, (2,{6},6) => 1, (3,{5},5) => 6, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (1,{1},1) => 2, (1,{3},3) => 21, (2,{4},4) => 24, (2,{6},6) => 4, (3,{5},5) --------------------------------------------------------------------------------------------- => 5, (3,{7},7) => 7, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{1},1) => 3, (1,{3},3) => 24, (2,{4},4) => 25, (2,{6},6) => 1, (3,{5},5) --------------------------------------------------------------------------------------------- => 4, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{1},1) => 4, (1,{3},3) => 19, (2,{4},4) => 22, (2,{6},6) => 4, (3,{5},5) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 9, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 3, --------------------------------------------------------------------------------------------- (1,{1},1) => 5, (1,{3},3) => 22, (2,{4},4) => 23, (2,{6},6) => 1, (3,{5},5) => 2, (3,{7},7) --------------------------------------------------------------------------------------------- => 6, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (1,{1},1) => 6, --------------------------------------------------------------------------------------------- (1,{3},3) => 17, (2,{4},4) => 20, (2,{6},6) => 4, (3,{5},5) => 1, (3,{7},7) => 11, (4,{8},8) --------------------------------------------------------------------------------------------- => 4}, new BettiTally from {(0,{0},0) => 5, (0,{1},1) => 3, (1,{1},1) => 7, (1,{3},3) => 20, --------------------------------------------------------------------------------------------- (2,{4},4) => 23, (2,{6},6) => 3, (3,{7},7) => 8, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (1,{3},3) => 8, (1,{5},5) => 4, (2,{4},4) => 1, (2,{6},6) => 27, (3,{7},7) --------------------------------------------------------------------------------------------- => 20, (3,{9},9) => 3, (4,{8},8) => 1, (4,{9},9) => 3, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (1,{3},3) => 8, (1,{5},5) => 7, (2,{4},4) => 1, (2,{5},5) => 3, (2,{6},6) => --------------------------------------------------------------------------------------------- 27, (3,{7},7) => 20, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 4, (1,{3},3) => 11, (1,{5},5) => 7, (2,{4},4) => 2, (2,{5},5) => 2, (2,{6},6) => --------------------------------------------------------------------------------------------- 24, (3,{7},7) => 17, (4,{8},8) => 1, (4,{9},9) => 1}, new BettiTally from {(0,{2},2) => 9, --------------------------------------------------------------------------------------------- (1,{3},3) => 16, (1,{5},5) => 7, (2,{4},4) => 3, (2,{5},5) => 3, (2,{6},6) => 19, (3,{7},7) --------------------------------------------------------------------------------------------- => 12, (4,{8},8) => 1}, new BettiTally from {(0,{2},2) => 11, (1,{3},3) => 20, (1,{4},4) => --------------------------------------------------------------------------------------------- 3, (2,{4},4) => 6, (2,{6},6) => 15, (3,{7},7) => 8, (3,{9},9) => 4, (4,{8},8) => 1, --------------------------------------------------------------------------------------------- (4,{10},10) => 2}, new BettiTally from {(0,{2},2) => 9, (1,{3},3) => 15, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (1,{5},5) => 1, (2,{4},4) => 3, (2,{6},6) => 18, (3,{7},7) => 13, (3,{9},9) => 3, (4,{8},8) --------------------------------------------------------------------------------------------- => 4, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 10, (1,{3},3) => 18, (1,{4},4) => --------------------------------------------------------------------------------------------- 3, (1,{5},5) => 2, (2,{4},4) => 6, (2,{6},6) => 17, (3,{7},7) => 10, (3,{9},9) => 2, --------------------------------------------------------------------------------------------- (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 4, (1,{3},3) => 7, --------------------------------------------------------------------------------------------- (1,{4},4) => 1, (1,{5},5) => 3, (2,{6},6) => 23, (3,{7},7) => 21, (3,{8},8) => 2, (3,{9},9) --------------------------------------------------------------------------------------------- => 1, (4,{8},8) => 7, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 4, (1,{3},3) => 7, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) => 2, (2,{6},6) => 23, (3,{7},7) => 21, (3,{9},9) --------------------------------------------------------------------------------------------- => 1, (4,{8},8) => 5, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 8, (1,{3},3) => --------------------------------------------------------------------------------------------- 13, (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) => 3, (2,{6},6) => 20, (3,{7},7) => 15, --------------------------------------------------------------------------------------------- (3,{9},9) => 1, (4,{8},8) => 4}, new BettiTally from {(0,{2},2) => 9, (1,{3},3) => 16, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 4, (2,{4},4) => 6, (2,{6},6) => 19, (3,{7},7) => 12, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 18, (2,{4},4) => 22, --------------------------------------------------------------------------------------------- (2,{6},6) => 5, (3,{5},5) => 4, (3,{6},6) => 3, (3,{7},7) => 10, (4,{8},8) => 5}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 4, (1,{3},3) => 26, (2,{4},4) => 26, (2,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{5},5) => 4, (3,{6},6) => 3, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{1},1) => 1, (1,{3},3) => 21, (2,{4},4) => 23, (2,{6},6) => 6, (3,{5},5) --------------------------------------------------------------------------------------------- => 3, (3,{6},6) => 3, (3,{7},7) => 7, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 4, (1,{1},1) => 2, (1,{3},3) => 24, (2,{4},4) => 24, (2,{6},6) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{6},6) => 3, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 4, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{1},1) => 3, (1,{3},3) => 19, (2,{4},4) => 21, (2,{6},6) => 6, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 9, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 7, (1,{1},1) => 3, (1,{3},3) => 27, (2,{4},4) => 25, (3,{5},5) => 1, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 1}, new BettiTally from {(0,{0},0) => 4, (0,{2},2) => 4, (1,{1},1) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 22, (2,{4},4) => 22, (2,{6},6) => 3, (3,{6},6) => 3, (3,{7},7) => 6, (4,{8},8) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 7, (1,{2},2) => 3, (1,{3},3) => 26, --------------------------------------------------------------------------------------------- (2,{4},4) => 26, (3,{5},5) => 4, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 7, (1,{1},1) => 2, (1,{2},2) => 3, (1,{3},3) => 24, (2,{4},4) => 24, (3,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 4, (0,{2},2) => 7, --------------------------------------------------------------------------------------------- (1,{1},1) => 4, (1,{2},2) => 3, (1,{3},3) => 22, (2,{4},4) => 22, (3,{7},7) => 6, (4,{8},8) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 18, (2,{4},4) => 22, --------------------------------------------------------------------------------------------- (2,{5},5) => 3, (2,{6},6) => 2, (3,{5},5) => 7, (3,{7},7) => 10, (4,{8},8) => 5}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{1},1) => 1, (0,{2},2) => 1, (1,{3},3) => 23, (2,{4},4) --------------------------------------------------------------------------------------------- => 25, (2,{5},5) => 2, (2,{6},6) => 3, (3,{5},5) => 7, (3,{7},7) => 5, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{1},1) => 3, (0,{2},2) => 1, (1,{3},3) => 25, (2,{4},4) --------------------------------------------------------------------------------------------- => 27, (2,{6},6) => 3, (3,{5},5) => 7, (3,{7},7) => 3}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 4, (1,{3},3) => 26, (2,{4},4) => 26, (2,{5},5) => 3, (3,{5},5) => 7, (3,{7},7) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 23, (2,{4},4) => 25, (2,{6},6) => 3, (3,{5},5) => 5, (3,{7},7) => 5, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, (0,{2},2) => 4, (1,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 26, (2,{4},4) => 26, (3,{5},5) => 4, (3,{7},7) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 4, (1,{3},3) => 21, (2,{4},4) --------------------------------------------------------------------------------------------- => 23, (2,{6},6) => 3, (3,{5},5) => 3, (3,{7},7) => 7, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 4, (1,{1},1) => 5, (1,{3},3) => 24, (2,{4},4) --------------------------------------------------------------------------------------------- => 24, (3,{5},5) => 2, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 4, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 6, (1,{3},3) => 19, (2,{4},4) => 21, (2,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{5},5) => 1, (3,{7},7) => 9, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{3},3) => 10, (1,{5},5) => 4, (2,{4},4) => 2, (2,{6},6) => 26, (3,{7},7) --------------------------------------------------------------------------------------------- => 18, (3,{9},9) => 3, (4,{9},9) => 3, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => --------------------------------------------------------------------------------------------- 2, (1,{3},3) => 8, (1,{5},5) => 6, (2,{4},4) => 2, (2,{6},6) => 28, (3,{7},7) => 20, --------------------------------------------------------------------------------------------- (3,{9},9) => 1, (4,{9},9) => 3}, new BettiTally from {(0,{0},0) => 2, (1,{3},3) => 8, --------------------------------------------------------------------------------------------- (1,{5},5) => 7, (2,{4},4) => 2, (2,{5},5) => 1, (2,{6},6) => 28, (3,{7},7) => 20, (4,{9},9) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 10, (1,{5},5) => 7, (2,{4},4) => 2, --------------------------------------------------------------------------------------------- (2,{5},5) => 3, (2,{6},6) => 22, (3,{7},7) => 18, (4,{8},8) => 2, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 4, (1,{3},3) => 10, (1,{5},5) => 7, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 24, (3,{7},7) => 18, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{3},3) => 10, (1,{5},5) => 7, (2,{4},4) => 2, (2,{5},5) => 3, (2,{6},6) => --------------------------------------------------------------------------------------------- 26, (3,{7},7) => 18, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 5, --------------------------------------------------------------------------------------------- (1,{3},3) => 13, (1,{5},5) => 7, (2,{4},4) => 3, (2,{5},5) => 2, (2,{6},6) => 23, (3,{7},7) --------------------------------------------------------------------------------------------- => 15, (4,{9},9) => 1}, new BettiTally from {(0,{2},2) => 10, (1,{3},3) => 18, (1,{5},5) => --------------------------------------------------------------------------------------------- 7, (2,{4},4) => 4, (2,{5},5) => 3, (2,{6},6) => 18, (3,{7},7) => 10}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 12, (1,{3},3) => 22, (1,{4},4) => 3, (2,{4},4) => 7, (2,{6},6) => 14, (3,{7},7) --------------------------------------------------------------------------------------------- => 6, (3,{9},9) => 4, (4,{10},10) => 2}, new BettiTally from {(0,{2},2) => 10, (1,{3},3) => --------------------------------------------------------------------------------------------- 17, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 4, (2,{6},6) => 17, (3,{7},7) => 11, --------------------------------------------------------------------------------------------- (3,{9},9) => 3, (4,{8},8) => 3, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 11, --------------------------------------------------------------------------------------------- (1,{3},3) => 20, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 7, (2,{6},6) => 16, (3,{7},7) --------------------------------------------------------------------------------------------- => 8, (3,{9},9) => 2, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 9, (1,{3},3) => --------------------------------------------------------------------------------------------- 15, (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) => 4, (2,{6},6) => 19, (3,{7},7) => 13, --------------------------------------------------------------------------------------------- (3,{9},9) => 1, (4,{8},8) => 3}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 10, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 4, (2,{4},4) => 5, (2,{6},6) => 22, (3,{7},7) => 18, (4,{8},8) --------------------------------------------------------------------------------------------- => 2, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 10, (1,{3},3) => 18, (1,{4},4) => --------------------------------------------------------------------------------------------- 3, (1,{5},5) => 4, (2,{4},4) => 7, (2,{6},6) => 18, (3,{7},7) => 10}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{2},2) => 2, (1,{1},1) => 1, (1,{3},3) => 23, (2,{4},4) => 24, (2,{6},6) --------------------------------------------------------------------------------------------- => 5, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) => 5, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{2},2) => 5, (1,{1},1) => 2, (1,{3},3) => 26, (2,{4},4) => 25, (2,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 5, (1,{1},1) => 3, (1,{3},3) => 19, (2,{4},4) => 21, (2,{6},6) => 2, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 9, (4,{8},8) => 5}, new BettiTally from {(0,{0},0) => 4, --------------------------------------------------------------------------------------------- (0,{2},2) => 2, (1,{1},1) => 3, (1,{3},3) => 21, (2,{4},4) => 22, (2,{6},6) => 5, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 7, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 4, --------------------------------------------------------------------------------------------- (0,{2},2) => 5, (1,{1},1) => 4, (1,{3},3) => 24, (2,{4},4) => 23, (2,{6},6) => 2, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 7, --------------------------------------------------------------------------------------------- (1,{1},1) => 2, (1,{2},2) => 2, (1,{3},3) => 26, (2,{4},4) => 25, (3,{5},5) => 2, (3,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 4, (0,{2},2) => 7, (1,{1},1) => 4, --------------------------------------------------------------------------------------------- (1,{2},2) => 2, (1,{3},3) => 24, (2,{4},4) => 23, (3,{6},6) => 1, (3,{7},7) => 4, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 1, (0,{2},2) => 2, (1,{3},3) => 25, --------------------------------------------------------------------------------------------- (2,{4},4) => 26, (2,{5},5) => 2, (2,{6},6) => 2, (3,{5},5) => 7, (3,{7},7) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 2, (1,{3},3) => --------------------------------------------------------------------------------------------- 25, (2,{4},4) => 26, (2,{6},6) => 2, (3,{5},5) => 5, (3,{7},7) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 4, (1,{3},3) => 23, (2,{4},4) --------------------------------------------------------------------------------------------- => 24, (2,{6},6) => 2, (3,{5},5) => 3, (3,{7},7) => 5, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 4, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 6, (1,{3},3) => 21, (2,{4},4) --------------------------------------------------------------------------------------------- => 22, (2,{6},6) => 2, (3,{5},5) => 1, (3,{7},7) => 7, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (1,{3},3) => 7, (1,{5},5) => 7, (2,{5},5) => 2, (2,{6},6) => 28, (3,{7},7) --------------------------------------------------------------------------------------------- => 21, (4,{8},8) => 2, (4,{9},9) => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 5, --------------------------------------------------------------------------------------------- (1,{3},3) => 12, (1,{5},5) => 7, (2,{4},4) => 1, (2,{5},5) => 3, (2,{6},6) => 23, (3,{7},7) --------------------------------------------------------------------------------------------- => 16, (4,{8},8) => 2}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 9, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (1,{5},5) => 1, (2,{4},4) => 2, (2,{6},6) => 21, (3,{7},7) => 19, (3,{9},9) => 3, (4,{8},8) --------------------------------------------------------------------------------------------- => 5, (4,{10},10) => 2}, new BettiTally from {(0,{2},2) => 11, (1,{3},3) => 19, (1,{4},4) => --------------------------------------------------------------------------------------------- 3, (1,{5},5) => 1, (2,{4},4) => 5, (2,{6},6) => 16, (3,{7},7) => 9, (3,{9},9) => 3, (4,{8},8) --------------------------------------------------------------------------------------------- => 2, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => 10, (1,{3},3) => 17, (1,{4},4) => --------------------------------------------------------------------------------------------- 3, (1,{5},5) => 3, (2,{4},4) => 5, (2,{6},6) => 18, (3,{7},7) => 11, (3,{9},9) => 1, --------------------------------------------------------------------------------------------- (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 3, (1,{3},3) => 22, (2,{4},4) => 24, --------------------------------------------------------------------------------------------- (2,{6},6) => 7, (3,{5},5) => 4, (3,{6},6) => 3, (3,{7},7) => 6, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 3, (0,{2},2) => 3, (1,{1},1) => 1, (1,{3},3) => 25, (2,{4},4) --------------------------------------------------------------------------------------------- => 25, (2,{6},6) => 4, (3,{5},5) => 3, (3,{6},6) => 3, (3,{7},7) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 2, (1,{1},1) => 2, (1,{3},3) => 16, (2,{4},4) => 20, (2,{6},6) --------------------------------------------------------------------------------------------- => 5, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) => 12, (4,{8},8) => 6}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 4, (1,{1},1) => 2, (1,{3},3) => 20, (2,{4},4) => 22, (2,{6},6) => 7, (3,{5},5) --------------------------------------------------------------------------------------------- => 2, (3,{6},6) => 3, (3,{7},7) => 8, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 4, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{1},1) => 3, (1,{3},3) => 23, (2,{4},4) => 23, (2,{6},6) => 4, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 5, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 5, --------------------------------------------------------------------------------------------- (1,{1},1) => 4, (1,{3},3) => 18, (2,{4},4) => 20, (2,{6},6) => 7, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 10, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 4, (0,{2},2) => 6, (1,{1},1) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 26, (2,{4},4) => 24, (2,{6},6) => 1, (3,{6},6) => 3, (3,{7},7) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 4, (0,{2},2) => 7, (1,{1},1) => 4, (1,{2},2) => 1, (1,{3},3) => --------------------------------------------------------------------------------------------- 26, (2,{4},4) => 24, (3,{6},6) => 2, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 22, (2,{4},4) => 24, (2,{5},5) => 3, (2,{6},6) => 4, (3,{5},5) => 7, (3,{7},7) --------------------------------------------------------------------------------------------- => 6, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 2, (1,{3},3) => 24, --------------------------------------------------------------------------------------------- (2,{4},4) => 26, (2,{5},5) => 1, (2,{6},6) => 4, (3,{5},5) => 7, (3,{7},7) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{1},1) => 3, (1,{1},1) => 1, (1,{3},3) => 24, (2,{4},4) --------------------------------------------------------------------------------------------- => 26, (2,{6},6) => 4, (3,{5},5) => 6, (3,{7},7) => 4}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (1,{1},1) => 3, (1,{3},3) => 22, (2,{4},4) => 24, (2,{6},6) => 4, (3,{5},5) --------------------------------------------------------------------------------------------- => 4, (3,{7},7) => 6, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{1},1) => 4, (1,{3},3) => 25, (2,{4},4) => 25, (2,{6},6) => 1, (3,{5},5) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 3}, new BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (1,{1},1) => 5, --------------------------------------------------------------------------------------------- (1,{3},3) => 20, (2,{4},4) => 22, (2,{6},6) => 4, (3,{5},5) => 2, (3,{7},7) => 8, (4,{8},8) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (0,{2},2) => 3, (1,{1},1) => 6, --------------------------------------------------------------------------------------------- (1,{3},3) => 23, (2,{4},4) => 23, (2,{6},6) => 1, (3,{5},5) => 1, (3,{7},7) => 5, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 5, (0,{1},1) => 3, (1,{1},1) => 7, (1,{3},3) => 18, --------------------------------------------------------------------------------------------- (2,{4},4) => 20, (2,{6},6) => 4, (3,{7},7) => 10, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 1, (1,{3},3) => 9, (1,{5},5) => 7, (2,{4},4) => 1, (2,{5},5) => --------------------------------------------------------------------------------------------- 2, (2,{6},6) => 27, (3,{7},7) => 19, (4,{8},8) => 1, (4,{9},9) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 6, (1,{3},3) => 14, (1,{5},5) => 7, (2,{4},4) => 2, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (2,{6},6) => 22, (3,{7},7) => 14, (4,{8},8) => 1}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 12, (1,{3},3) => 21, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 6, (2,{6},6) => 15, --------------------------------------------------------------------------------------------- (3,{7},7) => 7, (3,{9},9) => 3, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 6, (1,{3},3) => 12, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 5, (2,{6},6) --------------------------------------------------------------------------------------------- => 20, (3,{7},7) => 16, (3,{9},9) => 2, (4,{8},8) => 2, (4,{10},10) => 2}, new BettiTally --------------------------------------------------------------------------------------------- from {(0,{2},2) => 10, (1,{3},3) => 16, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 3, --------------------------------------------------------------------------------------------- (2,{6},6) => 18, (3,{7},7) => 12, (3,{9},9) => 2, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 11, (1,{3},3) => 19, (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) => 6, (2,{6},6) --------------------------------------------------------------------------------------------- => 17, (3,{7},7) => 9, (3,{9},9) => 1, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{3},3) => 24, (2,{4},4) => 25, (2,{6},6) => 6, (3,{5},5) => 4, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 4}, new BettiTally from {(0,{0},0) => 4, (0,{2},2) => 1, (1,{1},1) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 22, (2,{4},4) => 23, (2,{6},6) => 6, (3,{5},5) => 2, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 6, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 4, (0,{2},2) => 4, (1,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 25, (2,{4},4) => 24, (2,{6},6) => 3, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 3}, new BettiTally from {(0,{0},0) => 5, (0,{2},2) => 1, (1,{1},1) => 4, (1,{3},3) => 20, --------------------------------------------------------------------------------------------- (2,{4},4) => 21, (2,{6},6) => 6, (3,{6},6) => 3, (3,{7},7) => 8, (4,{8},8) => 2}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 4, (0,{2},2) => 7, (1,{1},1) => 3, (1,{2},2) => 3, (1,{3},3) => --------------------------------------------------------------------------------------------- 25, (2,{4},4) => 24, (3,{5},5) => 1, (3,{7},7) => 3}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{3},3) => 24, (2,{4},4) => 25, (2,{5},5) => 3, (2,{6},6) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 7, (3,{7},7) => 4}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 3, (0,{2},2) => 1, --------------------------------------------------------------------------------------------- (1,{1},1) => 3, (1,{3},3) => 24, (2,{4},4) => 25, (2,{6},6) => 3, (3,{5},5) => 4, (3,{7},7) --------------------------------------------------------------------------------------------- => 4}, new BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 5, --------------------------------------------------------------------------------------------- (1,{3},3) => 22, (2,{4},4) => 23, (2,{6},6) => 3, (3,{5},5) => 2, (3,{7},7) => 6, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (0,{2},2) => 4, (1,{1},1) => 6, --------------------------------------------------------------------------------------------- (1,{3},3) => 25, (2,{4},4) => 24, (3,{5},5) => 1, (3,{7},7) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 5, (0,{1},1) => 3, (0,{2},2) => 1, (1,{1},1) => 7, (1,{3},3) => 20, (2,{4},4) --------------------------------------------------------------------------------------------- => 21, (2,{6},6) => 3, (3,{7},7) => 8, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 2, (1,{3},3) => 11, (1,{5},5) => 5, (2,{4},4) => 2, (2,{6},6) => 26, (3,{7},7) --------------------------------------------------------------------------------------------- => 17, (3,{9},9) => 2, (4,{9},9) => 3}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 11, (1,{5},5) => 7, (2,{4},4) => 2, (2,{5},5) => 2, (2,{6},6) => 26, (3,{7},7) --------------------------------------------------------------------------------------------- => 17, (4,{9},9) => 1}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 7, (1,{3},3) => 16, --------------------------------------------------------------------------------------------- (1,{5},5) => 7, (2,{4},4) => 3, (2,{5},5) => 3, (2,{6},6) => 21, (3,{7},7) => 12}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 13, (1,{3},3) => 23, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) --------------------------------------------------------------------------------------------- => 7, (2,{6},6) => 14, (3,{7},7) => 5, (3,{9},9) => 3, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 11, (1,{3},3) => 18, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 4, (2,{6},6) --------------------------------------------------------------------------------------------- => 17, (3,{7},7) => 10, (3,{9},9) => 2, (4,{8},8) => 3}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 12, (1,{3},3) => 21, (1,{4},4) => 3, (1,{5},5) => 3, (2,{4},4) => 7, (2,{6},6) => 16, --------------------------------------------------------------------------------------------- (3,{7},7) => 7, (3,{9},9) => 1}, new BettiTally from {(0,{0},0) => 4, (0,{2},2) => 2, --------------------------------------------------------------------------------------------- (1,{1},1) => 2, (1,{3},3) => 24, (2,{4},4) => 24, (2,{6},6) => 5, (3,{5},5) => 2, (3,{6},6) --------------------------------------------------------------------------------------------- => 3, (3,{7},7) => 4}, new BettiTally from {(0,{0},0) => 3, (0,{2},2) => 2, (1,{1},1) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 14, (2,{4},4) => 18, (2,{6},6) => 5, (3,{6},6) => 3, (3,{7},7) => 14, (4,{8},8) --------------------------------------------------------------------------------------------- => 7}, new BettiTally from {(0,{0},0) => 5, (0,{2},2) => 2, (1,{1},1) => 4, (1,{3},3) => 22, --------------------------------------------------------------------------------------------- (2,{4},4) => 22, (2,{6},6) => 5, (3,{6},6) => 3, (3,{7},7) => 6, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 5, (1,{3},3) => --------------------------------------------------------------------------------------------- 24, (2,{4},4) => 24, (2,{6},6) => 2, (3,{5},5) => 2, (3,{7},7) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 5, (0,{1},1) => 3, (0,{2},2) => 2, (1,{1},1) => 7, (1,{3},3) => 22, (2,{4},4) --------------------------------------------------------------------------------------------- => 22, (2,{6},6) => 2, (3,{7},7) => 6, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 2, (1,{3},3) => 10, (1,{5},5) => 7, (2,{5},5) => 3, (2,{6},6) => 26, (3,{7},7) --------------------------------------------------------------------------------------------- => 18, (4,{8},8) => 2}, new BettiTally from {(0,{2},2) => 7, (1,{3},3) => 14, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (2,{4},4) => 5, (2,{6},6) => 18, (3,{7},7) => 14, (3,{9},9) => 4, (4,{8},8) => 2, (4,{10},10) --------------------------------------------------------------------------------------------- => 3}, new BettiTally from {(0,{2},2) => 13, (1,{3},3) => 22, (1,{4},4) => 3, (2,{4},4) => 5, --------------------------------------------------------------------------------------------- (2,{6},6) => 14, (3,{7},7) => 6, (3,{9},9) => 4, (4,{8},8) => 2, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 12, (1,{3},3) => 20, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) --------------------------------------------------------------------------------------------- => 5, (2,{6},6) => 16, (3,{7},7) => 8, (3,{9},9) => 2, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 4, (1,{1},1) => 1, (1,{3},3) => 23, (2,{4},4) => 24, (2,{6},6) => 7, (3,{5},5) --------------------------------------------------------------------------------------------- => 3, (3,{6},6) => 3, (3,{7},7) => 5}, new BettiTally from {(0,{0},0) => 5, (1,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 21, (2,{4},4) => 22, (2,{6},6) => 7, (3,{5},5) => 1, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 7, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 5, (0,{2},2) => 3, (1,{1},1) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 24, (2,{4},4) => 23, (2,{6},6) => 4, (3,{6},6) => 3, (3,{7},7) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{1},1) => 1, (0,{2},2) => 2, (1,{3},3) => 23, (2,{4},4) --------------------------------------------------------------------------------------------- => 27, (2,{5},5) => 2, (2,{6},6) => 1, (3,{5},5) => 7, (3,{7},7) => 5, (4,{10},10) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 4, (0,{1},1) => 3, (1,{1},1) => 4, (1,{3},3) => 23, (2,{4},4) --------------------------------------------------------------------------------------------- => 24, (2,{6},6) => 4, (3,{5},5) => 3, (3,{7},7) => 5}, new BettiTally from {(0,{0},0) => 5, --------------------------------------------------------------------------------------------- (0,{1},1) => 3, (1,{1},1) => 6, (1,{3},3) => 21, (2,{4},4) => 22, (2,{6},6) => 4, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{7},7) => 7, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 5, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{1},1) => 7, (1,{3},3) => 24, (2,{4},4) => 23, (2,{6},6) => 1, (3,{7},7) --------------------------------------------------------------------------------------------- => 4}, new BettiTally from {(0,{0},0) => 1, (1,{3},3) => 5, (1,{5},5) => 7, (2,{4},4) => 1, --------------------------------------------------------------------------------------------- (2,{5},5) => 2, (2,{6},6) => 27, (3,{7},7) => 23, (4,{8},8) => 2, (4,{9},9) => 1, (4,{10},10) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 3, (1,{3},3) => 12, (1,{5},5) => 7, --------------------------------------------------------------------------------------------- (2,{4},4) => 1, (2,{5},5) => 3, (2,{6},6) => 25, (3,{7},7) => 16, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 14, (1,{3},3) => 24, (1,{4},4) => 3, (2,{4},4) => 6, (2,{6},6) --------------------------------------------------------------------------------------------- => 13, (3,{7},7) => 4, (3,{9},9) => 4, (4,{8},8) => 1, (4,{10},10) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 13, (1,{3},3) => 22, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 6, (2,{6},6) --------------------------------------------------------------------------------------------- => 15, (3,{7},7) => 6, (3,{9},9) => 2, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 5, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{1},1) => 3, (1,{3},3) => 23, (2,{4},4) => 23, (2,{6},6) => 6, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 3, (3,{7},7) => 5}, new BettiTally from {(0,{0},0) => 5, (0,{1},1) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{1},1) => 6, (1,{3},3) => 23, (2,{4},4) => 23, (2,{6},6) => 3, (3,{5},5) --------------------------------------------------------------------------------------------- => 1, (3,{7},7) => 5}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 4, (1,{3},3) => 14, --------------------------------------------------------------------------------------------- (1,{5},5) => 7, (2,{4},4) => 2, (2,{5},5) => 3, (2,{6},6) => 24, (3,{7},7) => 14}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{2},2) => 15, (1,{3},3) => 26, (1,{4},4) => 3, (2,{4},4) => 7, (2,{6},6) --------------------------------------------------------------------------------------------- => 12, (3,{7},7) => 2, (3,{9},9) => 4, (4,{10},10) => 1}, new BettiTally from {(0,{2},2) => --------------------------------------------------------------------------------------------- 14, (1,{3},3) => 24, (1,{4},4) => 3, (1,{5},5) => 2, (2,{4},4) => 7, (2,{6},6) => 14, --------------------------------------------------------------------------------------------- (3,{7},7) => 4, (3,{9},9) => 2}, new BettiTally from {(0,{2},2) => 14, (1,{3},3) => 23, --------------------------------------------------------------------------------------------- (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 5, (2,{6},6) => 14, (3,{7},7) => 5, (3,{9},9) => --------------------------------------------------------------------------------------------- 3, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 6, (1,{1},1) => 4, (1,{3},3) => 22, --------------------------------------------------------------------------------------------- (2,{4},4) => 22, (2,{6},6) => 7, (3,{6},6) => 3, (3,{7},7) => 6}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 6, (0,{1},1) => 3, (1,{1},1) => 7, (1,{3},3) => 22, (2,{4},4) => 22, (2,{6},6) --------------------------------------------------------------------------------------------- => 4, (3,{7},7) => 6}, new BettiTally from {(0,{0},0) => 3, (1,{3},3) => 10, (1,{5},5) => 7, --------------------------------------------------------------------------------------------- (2,{5},5) => 3, (2,{6},6) => 28, (3,{7},7) => 18, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 15, (1,{3},3) => 25, (1,{4},4) => 3, (1,{5},5) => 1, (2,{4},4) => 6, (2,{6},6) --------------------------------------------------------------------------------------------- => 13, (3,{7},7) => 3, (3,{9},9) => 3, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 3, --------------------------------------------------------------------------------------------- (0,{2},2) => 1, (1,{3},3) => 12, (1,{5},5) => 7, (2,{4},4) => 1, (2,{5},5) => 3, (2,{6},6) => --------------------------------------------------------------------------------------------- 27, (3,{7},7) => 16}, new BettiTally from {(0,{2},2) => 16, (1,{3},3) => 27, (1,{4},4) => 3, --------------------------------------------------------------------------------------------- (1,{5},5) => 1, (2,{4},4) => 7, (2,{6},6) => 12, (3,{7},7) => 1, (3,{9},9) => 3}}; ---------------------- if g == 20 then return {new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 2, (1,{3},3) => 16, (2,{4},4) => 17, --------------------------------------------------------------------------------------------- (2,{6},6) => 4, (3,{6},6) => 4, (3,{7},7) => 8, (4,{8},8) => 4}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{1},1) => 4, (1,{2},2) => 7, (1,{4},4) => 6, (2,{4},4) => 2, (2,{5},5) => 20, (3,{6},6) --------------------------------------------------------------------------------------------- => 15, (4,{8},8) => 2}, new BettiTally from {(0,{1},1) => 5, (1,{2},2) => 9, (1,{4},4) => 6, --------------------------------------------------------------------------------------------- (2,{4},4) => 4, (2,{5},5) => 18, (3,{6},6) => 13, (4,{9},9) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 3, (1,{3},3) => 4, (1,{4},4) => 4, (2,{6},6) => 19, (3,{7},7) => 20, (4,{8},8) --------------------------------------------------------------------------------------------- => 6}, new BettiTally from {(0,{2},2) => 6, (1,{3},3) => 12, (1,{4},4) => 4, (2,{4},4) => 6, --------------------------------------------------------------------------------------------- (2,{6},6) => 15, (3,{7},7) => 12, (4,{10},10) => 1}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 3, (1,{3},3) => 18, (2,{4},4) => 18, (2,{6},6) => 3, (3,{6},6) => 4, (3,{7},7) --------------------------------------------------------------------------------------------- => 6, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 6, (1,{2},2) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 18, (2,{4},4) => 18, (3,{6},6) => 1, (3,{7},7) => 6, (4,{8},8) => 3}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{1},1) => 1, (1,{3},3) => 16, (2,{4},4) => 20, (2,{5},5) --------------------------------------------------------------------------------------------- => 3, (3,{5},5) => 6, (3,{7},7) => 6, (4,{8},8) => 3}, new BettiTally from {(0,{1},1) => 3, --------------------------------------------------------------------------------------------- (1,{3},3) => 18, (2,{4},4) => 22, (2,{5},5) => 1, (3,{5},5) => 6, (3,{7},7) => 4, (4,{8},8) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{2},2) => 4, (1,{3},3) => 6, (1,{4},4) => 3, (2,{6},6) => 18, --------------------------------------------------------------------------------------------- (3,{7},7) => 18, (3,{8},8) => 1, (4,{8},8) => 6}, new BettiTally from {(0,{2},2) => 4, --------------------------------------------------------------------------------------------- (1,{3},3) => 6, (1,{4},4) => 4, (2,{4},4) => 1, (2,{6},6) => 18, (3,{7},7) => 18, (4,{8},8) --------------------------------------------------------------------------------------------- => 5}, new BettiTally from {(0,{0},0) => 1, (0,{2},2) => 4, (1,{3},3) => 20, (2,{4},4) => 19, --------------------------------------------------------------------------------------------- (2,{6},6) => 2, (3,{6},6) => 4, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 6, (1,{2},2) => 2, (1,{3},3) => 20, (2,{4},4) => 19, (3,{6},6) --------------------------------------------------------------------------------------------- => 2, (3,{7},7) => 4, (4,{8},8) => 2}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 8, --------------------------------------------------------------------------------------------- (1,{4},4) => 4, (2,{4},4) => 2, (2,{6},6) => 17, (3,{7},7) => 16, (4,{8},8) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 1, (0,{2},2) => 5, (1,{3},3) => 22, (2,{4},4) => 20, (2,{6},6) --------------------------------------------------------------------------------------------- => 1, (3,{6},6) => 4, (3,{7},7) => 2, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 1, --------------------------------------------------------------------------------------------- (0,{2},2) => 6, (1,{2},2) => 1, (1,{3},3) => 22, (2,{4},4) => 20, (3,{6},6) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 2, (4,{8},8) => 1}, new BettiTally from {(0,{1},1) => 6, (1,{2},2) => 10, (1,{4},4) => 6, --------------------------------------------------------------------------------------------- (2,{4},4) => 3, (2,{5},5) => 18, (3,{6},6) => 12, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{2},2) => 6, (1,{3},3) => 10, (1,{4},4) => 4, (2,{4},4) => 3, (2,{6},6) => 16, (3,{7},7) --------------------------------------------------------------------------------------------- => 14, (4,{8},8) => 3}, new BettiTally from {(0,{0},0) => 2, (1,{3},3) => 16, (2,{4},4) => --------------------------------------------------------------------------------------------- 17, (2,{6},6) => 6, (3,{6},6) => 4, (3,{7},7) => 8, (4,{8},8) => 3}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 6, (1,{3},3) => 24, (2,{4},4) => 21, (3,{6},6) => 4}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (1,{3},3) => 17, (2,{4},4) => 20, (2,{5},5) => 4, (3,{5},5) --------------------------------------------------------------------------------------------- => 6, (3,{7},7) => 5, (4,{8},8) => 2}, new BettiTally from {(0,{0},0) => 1, (0,{1},1) => 2, --------------------------------------------------------------------------------------------- (1,{3},3) => 19, (2,{4},4) => 22, (2,{5},5) => 2, (3,{5},5) => 6, (3,{7},7) => 3, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{1},1) => 4, (1,{3},3) => 21, (2,{4},4) => 24, (3,{5},5) => 6, --------------------------------------------------------------------------------------------- (3,{7},7) => 1}, new BettiTally from {(0,{2},2) => 7, (1,{3},3) => 12, (1,{4},4) => 4, --------------------------------------------------------------------------------------------- (2,{4},4) => 4, (2,{6},6) => 15, (3,{7},7) => 12, (4,{8},8) => 2}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 2, (0,{2},2) => 1, (1,{3},3) => 18, (2,{4},4) => 18, (2,{6},6) => 5, (3,{6},6) --------------------------------------------------------------------------------------------- => 4, (3,{7},7) => 6, (4,{8},8) => 2}, new BettiTally from {(0,{2},2) => 8, (1,{3},3) => 14, --------------------------------------------------------------------------------------------- (1,{4},4) => 4, (2,{4},4) => 5, (2,{6},6) => 14, (3,{7},7) => 10, (4,{8},8) => 1}, new --------------------------------------------------------------------------------------------- BettiTally from {(0,{0},0) => 2, (0,{2},2) => 2, (1,{3},3) => 20, (2,{4},4) => 19, (2,{6},6) --------------------------------------------------------------------------------------------- => 4, (3,{6},6) => 4, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 6, (1,{2},2) => 4, (1,{3},3) => 20, (2,{4},4) => 19, (3,{7},7) => 4, (4,{8},8) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{1},1) => 8, (1,{2},2) => 13, (1,{4},4) => 6, (2,{4},4) => 4, --------------------------------------------------------------------------------------------- (2,{5},5) => 16, (3,{6},6) => 9}, new BettiTally from {(0,{2},2) => 9, (1,{3},3) => 16, --------------------------------------------------------------------------------------------- (1,{4},4) => 4, (2,{4},4) => 6, (2,{6},6) => 13, (3,{7},7) => 8}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 1, (0,{2},2) => 1, (1,{3},3) => 14, (2,{4},4) => 16, (2,{6},6) => 5, (3,{6},6) --------------------------------------------------------------------------------------------- => 4, (3,{7},7) => 10, (4,{8},8) => 5}, new BettiTally from {(0,{2},2) => 5, (1,{3},3) => 10, --------------------------------------------------------------------------------------------- (1,{4},4) => 4, (2,{4},4) => 5, (2,{6},6) => 16, (3,{7},7) => 14, (4,{8},8) => 1, (4,{10},10) --------------------------------------------------------------------------------------------- => 1}, new BettiTally from {(0,{0},0) => 2, (0,{2},2) => 3, (1,{3},3) => 22, (2,{4},4) => 20, --------------------------------------------------------------------------------------------- (2,{6},6) => 3, (3,{6},6) => 4, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{2},2) => 6, (1,{2},2) => 3, (1,{3},3) => 22, (2,{4},4) => 20, (3,{6},6) => 1, (3,{7},7) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 2, (0,{1},1) => 1, (1,{3},3) => 20, (2,{4},4) => 22, --------------------------------------------------------------------------------------------- (2,{5},5) => 3, (3,{5},5) => 6, (3,{7},7) => 2}, new BettiTally from {(0,{0},0) => 2, --------------------------------------------------------------------------------------------- (0,{1},1) => 4, (1,{1},1) => 3, (1,{3},3) => 20, (2,{4},4) => 22, (3,{5},5) => 3, (3,{7},7) --------------------------------------------------------------------------------------------- => 2}, new BettiTally from {(0,{0},0) => 3, (0,{1},1) => 4, (1,{1},1) => 5, (1,{3},3) => 18, --------------------------------------------------------------------------------------------- (2,{4},4) => 20, (3,{5},5) => 1, (3,{7},7) => 4, (4,{8},8) => 1}, new BettiTally from --------------------------------------------------------------------------------------------- {(0,{0},0) => 3, (1,{3},3) => 20, (2,{4},4) => 19, (2,{6},6) => 6, (3,{6},6) => 4, (3,{7},7) --------------------------------------------------------------------------------------------- => 4}, new BettiTally from {(0,{0},0) => 4, (0,{1},1) => 4, (1,{1},1) => 6, (1,{3},3) => 19, --------------------------------------------------------------------------------------------- (2,{4},4) => 20, (3,{7},7) => 3}}; ) TEST /// p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; beta=betti map(S^{22:0,6:-2},S^{0:-1,4:-2,24:-3},0) possMF=listOfPossibleMatFac(beta,4); time pcand20=candidateTables(beta,4); -- takes about 53 seconds cand20=select(pcand20,B->any(possMF,l->l==arisingMatFac(B,4))); cand20 == precompiledListOfCandidates(20) precompiledListOfCandidates(20) /// ------------------------------------------------------------------ -- Verify all the assertions of the paper -- ------------------------------------------------------------------ verifyAssertionsOfThePaper = method() verifyAssertionsOfThePaper(ZZ) := n -> ( if n == 1 then ( print " /// ------------------------------------------------------------------ -- Proposition 2.1 -- -- Expected Betti tables of a general curve of genus 12 and -- -- degree 14 -- ------------------------------------------------------------------ p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; -- homogeneous coordinate ring of P4 -- We construct a curve C of genus 12 and degree 14 using -- Algorithm 4.6 time IC=randomCurveGenus12Degree14InP4(S); -- takes 13 sec -- C is smooth of the right genus and degree (codim IC, genus IC, degree IC) == (3, 12, 14) isSmoothCurve(IC) -- C and its section module have expected Betti table betti res IC omegaC=Ext^2(IC,S^{ -5}); -- canonical module of C fomega=res omegaC; sM=S^{ -5}**coker transpose fomega.dd_3; betti res sM gIC=gens IC; X=ideal(gIC * random(source gIC,S^{-min(degrees source gIC)})); betti res prune(sM**(S/X)) /// "); -- if n == 2 then print " /// ------------------------------------------------------------------ -- Theorem 3.1, Remark 3.4, Corollary 3.5 -- -- For each pair (g,d), we construct a matrix factorization of -- -- the given shape giving rise to monads and smooth curves as -- -- claimed in Theorem 3.1 -- -- For (g,d)=(12,14) or (13,15) we show that a general matrix -- -- factorization has a component with kernel of the expected -- -- dimension, giving thus rise to monads parameterized by a -- -- rational variety. We show that the general monad produces a -- -- smooth curve of the right genus and degree, providing hence -- -- a proof of Remark 3.4 and Corollary 3.5 -- ------------------------------------------------------------------ -- p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; -- homogeneous coordinate ring of P4 -- ---------- (g,d) = (12,14) --------------------------------------- -- We construct a matrix factorization starting from a nodal -- curve, as explained in Corollary 3.5 time singC=singularCurveInP4(S,12,14); -- takes 4 sec -- We check that the curve is singular in P4: isSmoothCurve(singC) == false -- time codim(singularLocus(singC)) == 4 -- takes about 90 sec omegaSingC=Ext^2(singC,S^{ -5}); -- canonical module of C fomegaSing=res omegaSingC; sM=S^{ -5}**coker transpose fomegaSing.dd_3; (psi,phi)=matrixFactorizationFromModule(sM); X=ideal ring phi; codim X, degree X -- We cheack that (phi,psi) is a matrix factorization on the cubic -- threefold X of the correct shape: ker phi = image psi prune ((ker psi) / (image phi)) == 0 -- and phi is the presentation of a MCM module over S betti res (coker psi ** S) betti psi -- We check that the dimension of the kernel of the last row -- of the Betti table of psi is 5, i.e., the expected one betti syz((psi_{2..16})^{15,16},DegreeLimit=>2) -- We construct the curve C monadShape=betti map(S^{2:-1},S^{2:-1,2:-2},0); IC=idealFromMatFac(psi, monadShape); -- We check that C is a smooth curve of genus 12 and degree 14 -- of maximal rank. We check also that the Betti table of its -- section module is as expected over S and over the ring of -- a general supporting cubic threefold (codim IC, genus IC, degree IC) == (3, 12, 14) isSmoothCurve(IC) betti res IC -- expected Betti table omegaC=Ext^2(IC,S^{ -5}); -- canonical module of C fomega=res omegaC; sM=S^{ -5}**coker transpose fomega.dd_3; betti res sM gIE=gens IC; Y=ideal(gIE * random(source gIE,S^{-3})); betti res (sM**(S/Y)) -- non-minimal resolution betti res prune (sM**(S/Y)) -- expected resolution -- ---------- (g,d) = (13,15) --------------------------------------- -- We construct a matrix factorization starting from a nodal -- curve, as explained in Corollary 3.5 time singC=singularCurveInP4(S,13,15); -- takes 3 sec -- We check that the curve is singular in P4: isSmoothCurve(singC) == false -- time codim(singularLocus(singC)) == 4 -- takes about 379 sec omegaSingC=Ext^2(singC,S^{ -5}); -- canonical module of C fomegaSing=res omegaSingC; sM=S^{ -5}**coker transpose fomegaSing.dd_3; (psi,phi)=matrixFactorizationFromModule(sM); X=ideal ring phi; codim X, degree X -- We cheack that (phi,psi) is a matrix factorization on the cubic -- threefold X of the correct shape: ker phi = image psi prune ((ker psi) / (image phi)) == 0 -- and phi is the presentation of a MCM module over S betti res (coker psi ** S) betti psi -- We check that the dimension of the kernel of the last row -- of the Betti table of psi is 3, i.e., the expected one betti syz((psi_{3..20})^{18,19,20},DegreeLimit=>2) -- We construct the curve C monadShape=betti map(S^{3:-1},S^{3:-1,2:-2},0); IC=idealFromMatFac(psi, monadShape); -- We check that C is a smooth curve of genus 13 and degree 15 -- of maximal rank. We check also that the Betti table of its -- section module is as expected over S and over the ring of -- a general supporting cubic threefold (codim IC, genus IC, degree IC) == (3, 13, 15) isSmoothCurve(IC) betti res IC -- expected Betti table omegaC=Ext^2(IC,S^{1:-5}); -- canonical module of C fomega=res omegaC; sM=S^{ -5}**coker transpose fomega.dd_3; betti res sM gIE=gens IC; Y=ideal(gIE * random(source gIE,S^{1:-3})); betti res (sM**(S/Y)) -- non-minimal resolution betti res prune (sM**(S/Y)) -- expected resolution -- ---------- (g,d) = (16,17) --------------------------------------- -- We construct a matrix factorization starting from a curve on -- the Alexander surface Y, as explained in Section 5.2 time alexC=first curveOnAlexanderSurface(S,16,17); -- takes 23 sec omegaAlexC=Ext^2(alexC,S^{1:-5}); -- canonical module of C fomegaAlex=res omegaAlexC; sM=S^{1:-5}**coker transpose fomegaAlex.dd_3; (psi,phi)=matrixFactorizationFromModule(sM); X=ideal ring phi; codim X, degree X -- We cheack that (phi,psi) is a matrix factorization on the -- quartic threefold X of the correct shape: ker phi = image psi time prune ((ker psi) / (image phi)) == 0 -- takes 31 sec -- and phi is the presentation of a MCM module betti res (coker psi ** S) -- We construct the curve C monadShape=betti map(S^{4:-2},S^{3:-2,1:-1},0); time IC=idealFromMatFac(psi, monadShape); -- takes 434 sec -- We check that C is a smooth curve of genus 13 and degree 15 -- of maximal rank. We check also that the Betti table of its -- section module is as expected over S and over the ring of -- a general supporting cubic threefold (codim IC, genus IC, degree IC) == (3, 16, 17) isSmoothCurve(IC) betti res IC -- expected Betti table omegaC=Ext^2(IC,S^{1:-5}); -- canonical module of C fomega=res omegaC; sM=S^{1:-5}**coker transpose fomega.dd_3; betti res sM gIE=gens IC; Y=ideal(gIE * random(source gIE,S^{-4})); betti res (sM**(S/Y)) -- expected resolution -- ---------- (g,d) = (17,18) --------------------------------------- -- We construct a matrix factorization starting from a curve on -- the Alexander surface Y, as explained in Section 5.2 time alexC=first curveOnAlexanderSurface(S,17,18); -- takes 25 sec omegaAlexC=Ext^2(alexC,S^{-5}); -- canonical module of C fomegaAlex=res omegaAlexC; sM=S^{1:-5}**coker transpose fomegaAlex.dd_3; (psi,phi)=matrixFactorizationFromModule(sM); X=ideal ring phi; codim X, degree X -- We cheack that (phi,psi) is a matrix factorization on the -- quartic threefold X of the correct shape: ker phi = image psi time prune ((ker psi) / (image phi)) == 0 -- takes 148 sec -- and phi is the presentation of a MCM module betti res (coker psi ** S) -- We construct the curve C monadShape=betti map(S^{5:-2},S^{3:-2,2:-1},0); time IC=idealFromMatFac(psi, monadShape); -- takes 1134 sec -- We check that C is a smooth curve of genus 13 and degree 15 -- of maximal rank. We check also that the Betti table of its -- section module is as expected over S and over the ring of -- a general supporting cubic threefold (codim IC, genus IC, degree IC) == (3, 17, 18) isSmoothCurve(IC) betti res IC -- expected Betti table omegaC=Ext^2(IC,S^{-5}); -- canonical module of C fomega=res omegaC; sM=S^{1:-5}**coker transpose fomega.dd_3; betti res sM gIE=gens IC; Y=ideal(gIE * random(source gIE,S^{-4})); betti res (sM**(S/Y)) -- expected resolution -- ---------- (g,d) = (18,19) --------------------------------------- -- We construct a matrix factorization starting from a curve on -- the Alexander surface Y, as explained in Section 5.2 time alexC=first curveOnAlexanderSurface(S,18,19); -- takes 23 sec omegaAlexC=Ext^2(alexC,S^{-5}); -- canonical module of C fomegaAlex=res omegaAlexC; sM=S^{1:-5}**coker transpose fomegaAlex.dd_3; (psi,phi)=matrixFactorizationFromModule(sM); X=ideal ring phi; codim X, degree X -- We cheack that (phi,psi) is a matrix factorization on the -- quartic threefold X of the correct shape: ker phi = image psi time prune ((ker psi) / (image phi)) == 0 -- takes 437 sec -- and phi is the presentation of a MCM module betti res (coker psi ** S) -- We construct the curve C monadShape=betti map(S^{6:-2},S^{3:-2,3:-1},0); time IC=idealFromMatFac(psi, monadShape); -- takes 2227 sec -- We check that C is a smooth curve of genus 13 and degree 15 -- of maximal rank. We check also that the Betti table of its -- section module is as expected over S and over the ring of -- a general supporting cubic threefold (codim IC, genus IC, degree IC) == (3, 18, 19) isSmoothCurve(IC) betti res IC -- expected Betti table omegaC=Ext^2(IC,S^{-5}); -- canonical module of C fomega=res omegaC; sM=S^{1:-5}**coker transpose fomega.dd_3; betti res sM gIE=gens IC; Y=ideal(gIE * random(source gIE,S^{-4})); betti res (sM**(S/Y)) -- expected resolution -- ---------- (g,d) = (19,20) --------------------------------------- -- We construct a matrix factorization starting from a curve on -- the Alexander surface Y, as explained in Section 5.2 time alexC=first curveOnAlexanderSurface(S,19,20); -- takes 22 sec omegaAlexC=Ext^2(alexC,S^{-5}); -- canonical module of C fomegaAlex=res omegaAlexC; sM=S^{1:-5}**coker transpose fomegaAlex.dd_3; (psi,phi)=matrixFactorizationFromModule(sM); X=ideal ring phi; codim X, degree X -- We cheack that (phi,psi) is a matrix factorization on the -- quartic threefold X of the correct shape: ker phi = image psi time prune ((ker psi) / (image phi)) == 0 -- takes 1125 sec -- and phi is the presentation of a MCM module betti res (coker psi ** S) -- We construct the curve C monadShape=betti map(S^{7:-2},S^{3:-2,4:-1},0); time IC=idealFromMatFac(psi, monadShape); -- takes 4721 sec -- We check that C is a smooth curve of genus 13 and degree 15 -- of maximal rank. We check also that the Betti table of its -- section module is as expected over S and over the ring of -- a general supporting cubic threefold (codim IC, genus IC, degree IC) == (3, 19, 20) isSmoothCurve(IC) betti res IC -- expected Betti table omegaC=Ext^2(IC,S^{-5}); -- canonical module of C fomega=res omegaC; sM=S^{1:-5}**coker transpose fomega.dd_3; betti res sM gIE=gens IC; Y=ideal(gIE * random(source gIE,S^{-4})); betti res (sM**(S/Y)) -- expected resolution -- ---------- (g,d) = (20,20) --------------------------------------- -- We construct a matrix factorization starting from a curve on -- the Alexander surface Y, as explained in Section 5.2 time alexC=first curveOnAlexanderSurface(S,20,20); -- takes 22 sec omegaAlexC=Ext^2(alexC,S^{-5}); -- canonical module of C fomegaAlex=res omegaAlexC; sM=S^{1:-5}**coker transpose fomegaAlex.dd_3; (psi,phi)=matrixFactorizationFromModule(sM); X=ideal ring phi; codim X, degree X -- We cheack that (phi,psi) is a matrix factorization on the -- quartic threefold X of the correct shape: ker phi = image psi time prune ((ker psi) / (image phi)) == 0 -- takes 7 sec -- and phi is the presentation of a MCM module betti res (coker psi ** S) -- We construct the curve C monadShape=betti map(S^{6:-2},S^{4:-2},0); time IC=idealFromMatFac(psi, monadShape); -- takes 3522 sec -- We check that C is a smooth curve of genus 13 and degree 15 -- of maximal rank. We check also that the Betti table of its -- section module is as expected over S and over the ring of -- a general supporting cubic threefold (codim IC, genus IC, degree IC) == (3, 20, 20) isSmoothCurve(IC) betti res IC -- expected Betti table omegaC=Ext^2(IC,S^{-5}); -- canonical module of C fomega=res omegaC; sM=S^{1:-5}**coker transpose fomega.dd_3; betti res sM gIE=gens IC; Y=ideal(gIE * random(source gIE,S^{-4})); betti res (sM**(S/Y)) -- expected resolution /// "; -- if n == 3 then print " /// ------------------------------------------------------------------ -- Example 4.1 -- -- List of candidate Betti tables giving rise to matrix -- -- factorizations of shape 15 2 -- -- 2 15 -- ------------------------------------------------------------------ -- p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; beta=betti map(S^{15:0,2:-1,0:-2},S^{2:-1,15:-2,0:-3},0) d=3; -- The following method produces the list of all tables lying in -- the Boij-Soederberg cone giving rise to a matrix -- factorization of the desired shape, assuming that no -- cancellation occurs. To speed up computations, the functions -- looks for tables with projective dimension at most 4. -- The function automatically takes care of the transposed -- shape, as well as of the shape of the syzygy map and its -- transposed. -- -- time bCands=candidateTables(beta,d); -- takes 2 sec -- If we look for Betti tables of modules of projective -- dimension at 5, i.e., modules with the irrelevant ideal -- associated, we can run time bCandsIrr=candidateTablesWithIrrelevantIdealAssociated(beta,d); -- takes 28 sec -- -- It is reasonable to assume that all the cancellations which -- are possible will occur. As a general philosophy, matrix -- factorizations with the maximum number of cancellations -- are expected to be easier to construct. We remark that some -- of the cancellations cannot occur because of the Shamash -- construction. -- Therefore, we can filter the previous list by asking that -- the arising matrix factorization has the fixed shape 15 2 -- 2 15, -- supposing that all the cancellations which are possible do -- occur. -- possibleShapes=listOfPossibleMatFac(beta,d); bCandsMaxCan=select(bCands,B->( arMatFac:=arisingMatFac(B,d); any(possibleShapes,s->s==arMatFac))); #bCandsMaxCan -- -- We get a list of 13 candidate Betti tables for which no -- cancellation is tolerated. We cannot find the Betti table of -- the section module of a general curve of genus 12 and degree -- 14 as, in that specific case, we had one cancellation. -- -- In the same line of thought, we can look for candidate Betti -- tables whose induced non-minimal matrix factorization has -- shape 1 -- 15 2 -- 2 16 -- beta'=betti map(S^{1:1,15:0,2:-1,0:-2},S^{2:-1,16:-2,0:-3},0) time bCands'=candidateTables(beta',d); -- takes 9 sec -- -- and filter them according to the expected minimal matrix -- factorization they may give rise to, which has to coincide -- with 15 2 -- 2 15 -- bCandsMaxCan'=select(bCands',B->( arMatFac:=arisingMatFac(B,d); any(possibleShapes,s->s==arMatFac))); -- -- Among them, we find the Betti table of the section module -- of a general curve of genus 12 and degree 14 -- bCandsMaxCan'_9 -- -- Another possibility is to use the shape . 1 -- 16 2 -- 2 15 -- beta''=betti map(S^{0:1,16:0,2:-1,0:-2},S^{1:-0,2:-1,15:-2},0) time bCands''=candidateTables(beta'',d); -- takes 10 sec bCandsMaxCan''=select(bCands'',B->( arMatFac:=arisingMatFac(B,d); any(possibleShapes,s->s==arMatFac))); -- -- Among them, we find the Betti table exhibited in Example 4.1 -- bCandsMaxCan''_2 /// "; -- if n == 4 then print " /// ------------------------------------------------------------------ -- Proposition 4.2 -- -- A general curve of genus 10 and degree 13 has expected Betti -- -- table, and the matrix factorization it induces on a general -- -- supporting cubic threefold leads to a smooth curve of genus -- -- 12 and degree 14 -- ------------------------------------------------------------------ -- p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; -- We construct IE according to the first part of Algorithm 4.6 IE=randomCurveGenus10Degree13InP4(S); -- We check that IE is a smooth curve of genus 10 and degree 13 -- with the expected Betti table (codim IE, genus IE, degree IE) == (3,10,13) isSmoothCurve(IE) betti res IE -- We compute a matrix factorization on a supporting cubic -- threefold we construct a curve via the correspondence in the -- paper (phi,psi)=matrixFactorizationFromModule(IE); betti psi monadShape=betti map(S^{2:-1},S^{2:-1,2:-2},0); IC=idealFromMatFac(psi,monadShape); -- The curve we produced is smooth of the right genus and -- codimension (codim IC, genus IC, degree IC) == (3,12,14) isSmoothCurve(IC) betti res IC /// "; -- if n == 5 then print " /// ------------------------------------------------------------------ -- Proposition 4.4 -- -- A general matrix factorization of shape 15 2 -- -- 2 15 -- -- and a general choice of the quotient q yield a smooth curve -- -- of genus 10 and degree 13 -- ------------------------------------------------------------------ -- p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; time IC=randomCurveGenus12Degree14InP4(S); -- takes 11 sec omegaC=Ext^2(IC,S^{-5}); -- canonical module of C fomega=res omegaC; sM=S^{1:-5}**coker transpose fomega.dd_3; (psi,phi)=matrixFactorizationFromModule(sM); -- We choose a quotient q as in Proposition 4.4 SX=ring phi; A1=((psi_{0,1}))^{0..14}; A2=syz transpose A1; q=transpose(A2*random(source A2, SX^{4:0})); q*A1==0 -- q is a generally chosen quotient. We verify that the complex in -- Proposition 4.4 is a locally free resolution: incl=id_(source A1)||map(source q,source A1,0); alpha=inducedMap(target(q++id_(S^{2:-1})),coker phi,(q++id_(S^{2:-1}))*psi); beta=inducedMap(coker phi,source A1,incl); prune(ker(alpha)/image(beta)) == 0 -- exactness in F is verified prune ker(beta) == 0 -- exactness in O_X^2(-4) is verified too. -- We compute the ideal of the resulting curve with respect to SX -- and the ideal on S hom=Hom(coker ((q++id_(S^{2:-1}))*psi_{2..16}),SX); f=homomorphism hom_{0}; ICX = ann coker f; pr=map(SX,S); IC=preimage(pr,ICX); -- We check that IC is a smooth curve of genus 10 and degree 13 betti res IC (codim IC, genus IC, degree IC)==(3,10,13) isSmoothCurve(IC) /// "; -- if n == 6 then print " /// ------------------------------------------------------------------ -- Theorem 5.1, Section 5.2 -- -- Using auxiliary curves Z to construct curves C -- ------------------------------------------------------------------ -- p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; H={13,4,4,4,4,4,4,4,4,4,4}; -- the hyperplane class on Y (Y,pts)=alexanderSurface(S); -- we assume here that (g,d)=(16,17) Z={31,10,10,10,10,10,10,9,9,9,9} time IZ=properTransformAlexander(S,Z,pts); -- takes 25 sec N=auxiliarLineBundle(Y,IZ); betti res N -- the expected Betti table (phi,psi)=matrixFactorizationFromModule(N); betti dual psi monadShape=betti map(S^{4:-2},S^{3:-2,1:-1},0) time IC=idealFromMatFac(dual psi, monadShape); -- takes 358 sec -- We check that C has all the desired properties (codim IC, genus IC, degree IC) == (3, 16, 17) isSmoothCurve(IC) betti res IC -- expected Betti table omegaC=Ext^2(IC,S^{-5}); -- canonical module of C fomega=res omegaC; sM=S^{1:-5}**coker transpose fomega.dd_3; betti res sM gIC=gens IC; betti res (sM**(S/(ideal(gIC * random(source gIC,S^{-4}))))) -- expected resolution -- We check whether Y intersection X = C U Z X=ideal ring psi; YpX=Y+X; CiZ=intersect(IZ,IC); saturate YpX == CiZ -- true /// "; -- if n==7 then print " /// ------------------------------------------------------------------ -- Table 5.2 -- -- The linear systems reported in Table 5.2 yield curve of the -- -- desired genus and degree on the Alexander surface -- ------------------------------------------------------------------ -- p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; -- We construct the Alexander surface Y time (Y,pts)=alexanderSurface(S); -- takes 9 sec -- For every linear system listed in Table 5.2, we compute the -- proper transform on Y (g,d)=(16,17) L=linSysAlexander(g,d) time IC=properTransformAlexander(S,L,pts); -- takes 17 sec (codim IC, genus IC, degree IC) == (3,g,d) isSmoothCurve(IC) betti res IC -- (g,d)=(17,18) L=linSysAlexander(g,d) time IC=properTransformAlexander(S,L,pts); -- takes 19 sec (codim IC, genus IC, degree IC) == (3,g,d) isSmoothCurve(IC) betti res IC -- (g,d)=(18,19) L=linSysAlexander(g,d) time IC=properTransformAlexander(S,L,pts); -- takes 13 sec (codim IC, genus IC, degree IC) == (3,g,d) isSmoothCurve(IC) betti res IC -- (g,d)=(19,20) L=linSysAlexander(g,d) time IC=properTransformAlexander(S,L,pts); -- takes 15 sec (codim IC, genus IC, degree IC) == (3,g,d) isSmoothCurve(IC) betti res IC -- (g,d)=(20,20) L=linSysAlexander(g,d) time IC=properTransformAlexander(S,L,pts); -- takes 15 sec (codim IC, genus IC, degree IC) == (3,g,d) isSmoothCurve(IC) betti res IC /// "; if n == 8 then print " /// ------------------------------------------------------------------ -- Theorem 5.2 -- -- We compute h^1(N), where N is the normal sheaf of a curve C -- -- with respect to a general supporting threefold of minimal -- -- degree, and we check that it is zero for the general C -- ------------------------------------------------------------------ -- p=32009; -- a prime number Fp=ZZ/p; -- a prime field S=Fp[x_0..x_4]; -- (g,d)=(12,14) IC=randomCurveGenus12Degree14InP4(S); gIC=gens IC; X=ideal(gIC * random(source gIC,S^{-min(degrees source gIC)})); time h1NormalBundle(IC,X) == 0 -- takes 13 sec -- (g,d)=(13,15) IC=curveGenus13Degree15InP4(S); gIC=gens IC; X=ideal(gIC * random(source gIC,S^{-min(degrees source gIC)})); time h1NormalBundle(IC,X) == 0 -- takes 161 sec -- (g,d)=(16,17) (g,d)=(16,17); IC=first curveOnAlexanderSurface(S,g,d); gIC=gens IC; X=ideal(gIC * random(source gIC,S^{-min(degrees source gIC)})); time h1NormalBundle(IC,X) == 0 -- takes 1089 sec -- (g,d)=(17,18) (g,d)=(17,18); IC=first curveOnAlexanderSurface(S,g,d); gIC=gens IC; X=ideal(gIC * random(source gIC,S^{-min(degrees source gIC)})); time h1NormalBundle(IC,X) == 0 -- takes 1239 sec -- (g,d)=(18,19) (g,d)=(18,19); IC=first curveOnAlexanderSurface(S,g,d); gIC=gens IC; X=ideal(gIC * random(source gIC,S^{-min(degrees source gIC)})); time h1NormalBundle(IC,X) == 0 -- takes 435 sec -- (g,d)=(19,20) (g,d)=(19,20); IC=first curveOnAlexanderSurface(S,g,d); gIC=gens IC; X=ideal(gIC * random(source gIC,S^{-min(degrees source gIC)})); time h1NormalBundle(IC,X) == 0 -- takes 362 sec -- (g,d)=(20,20) (g,d)=(20,20); IC=first curveOnAlexanderSurface(S,g,d); gIC=gens IC; X=ideal(gIC * random(source gIC,S^{-min(degrees source gIC)})); time h1NormalBundle(IC,X) == 0 -- takes 242 sec /// "; if n==0 then ( verifyAssertionsOfThePaper(1), verifyAssertionsOfThePaper(2), verifyAssertionsOfThePaper(3), verifyAssertionsOfThePaper(4), verifyAssertionsOfThePaper(5), verifyAssertionsOfThePaper(6), verifyAssertionsOfThePaper(7), verifyAssertionsOfThePaper(8) ) ) TEST /// n = random(1,9) verifyAssertionsOfThePaper(n) /// beginDocumentation() document { Key => MatFacCurvesP4, Headline => "Matrix factorizations and curves in P^4", "This package implements the methods of the paper 'Matrix factorizations and curves in P^4' and serves as supporting code for the computational proofs it contains. The code needed to verify all the assertions of the paper is provided, as well as an output of its execution.", PARA{}, SUBSECTION "Main functions", UL{ TO "matrixFactorizationFromModule", TO "idealFromMatFac", TO "isSmoothCurve", TO "h1NormalBundle", TO "alexanderSurface", TO "linSysAlexander", TO "properTransformAlexander", TO "auxiliarLineBundle" }, PARA{}, SUBSECTION "Construction of curves", UL{ TO "singularCurveInP4", TO "curveGenus12Degree14InP4", TO "curveGenus13Degree15InP4", TO "randomCurveGenus10Degree13InP4", TO "randomCurveGenus12Degree14InP4", TO "randomGenus12Degree8CoverOfP1", TO "curveOnAlexanderSurface" }, PARA{}, SUBSECTION "Betti candidates", UL{ TO "isInBoijSoederbergCone", TO "listOfPossibleMatFac", TO "candidateTables", TO "candidateTablesWithIrrelevantIdealAssociated", TO "arisingMatFac", TO "precompiledListOfCandidates" }, PARA{}, SUBSECTION "Verify all the assertions of the paper", UL{ TO "verifyAssertionsOfThePaper" }, PARA{}, Caveat => {"This package requires Macaulay2 Version 1.8 or newer and the package ", {HREF{"https://www.math.uni-sb.de/ag/schreyer/images/data/computeralgebra/kGonalNodalCurves.m2", "kGonalNodalCurves.m2"}, " by Christian Bopp"}, ", which should be installed before this one." }, } doc /// Key matrixFactorizationFromModule (matrixFactorizationFromModule, Module) (matrixFactorizationFromModule, Ideal) Headline a matrix factorization induced by a quotient ring on a supporting hypersurface Usage matrixFactorizationFromModule M matrixFactorizationFromModule I Inputs M: Module over a ring of polynomials R I: Ideal in a ring of polynomials R Outputs : Sequence Description Text The function takes a random element f of minimal degree in the support of the annihilator of the module M (or in the ideal I) and produces the matrix factorization of f given by the periodic part of the R/f-resolution of M (or R/I) Example S = ZZ/32009[x_0..x_3]; I = minors(3,random(S^4,S^{1:-1,2:-2})); betti res I (phi, psi)=matrixFactorizationFromModule I; betti res ((S^1/I) ** (ring phi)) betti phi, betti psi SX = ring psi; phi*psi /// doc /// Key idealFromMatFac (idealFromMatFac, Matrix, BettiTally) Headline the ideal of a curve constructed from a matrix factorization Usage idealFromMatFac(psi,B) Inputs psi: Matrix one map of the matrix factorization B : BettiTally the shape of the monad Outputs : Ideal the ideal of a curve in the polynomial ring constructed from a random monad of the given shape Description Text The function takes the first map psi of a matrix factorization and the shape of the monad and realizes a random monad with such shape, from which the ideal of a curve is then constructed. The shape B should be a subtable of betti psi Example p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; time singC=singularCurveInP4(S,12,14); omegaSingC=Ext^2(singC,S^{ -5}); -- canonical module of C fomegaSing=res omegaSingC; sM=S^{ -5}**coker transpose fomegaSing.dd_3; (psi,phi)=matrixFactorizationFromModule(sM); betti psi monadShape=betti map(S^{2:-1},S^{2:-1,2:-2},0) IC=idealFromMatFac(psi, monadShape); betti res IC Caveat The function is implemented specifically for the cases of interest of the paper and might produce bizarre results, if any, in other non-verified cases. /// doc /// Key isSmoothCurve (isSmoothCurve,Ideal) Headline check whether an ideal defines a smooth curve Usage isSmoothCurve(C) Inputs C: Ideal the ideal of a curve in P^n, n >= 3 Outputs : Boolean true means that the curve is smooth, false means that the projected curve in P^3 is singular Description Text The function checks for smoothness with the Jacobian criterion. In a first step, we check for associated points. In case of n > 3 we project to P^3 from the center defined by the first 4 variables of the coordinate ring of P^n. Then we check for smoothness of the projected curve. We might be unlucky that the projection introduces a singularity as in the example of i17-o19 below Example kk=ZZ/101; S=kk[x_0..x_2]; C=ideal(x_0^2,x_0*x_1);isSmoothCurve C C=ideal(x_0*x_2^2-x_1^3);isSmoothCurve C S=kk[x_0..x_3]; C=intersect(ideal(x_0,x_1), ideal( x_1,x_2,x_3)) isSmoothCurve C C=ideal(x_0,x_1*x_2);isSmoothCurve C S=kk[x_0..x_4] C=ideal(x_0,x_1,x_2*x_3);isSmoothCurve C C=ideal(x_4,x_1,x_2*x_3);isSmoothCurve C m=matrix{{x_0,x_4,x_2,x_3},{x_4,x_2,x_3,x_1}} C=minors(2,m);isSmoothCurve C -- only the projected curve is singular C=minors(2,random(S^2,S^{4:-1}));isSmoothCurve C /// doc /// Key h1NormalBundle (h1NormalBundle, Ideal, Ideal) Headline computes h^1 of the normal sheaf Usage h1NormalBundle(I,X) Inputs I: Ideal the ideal of a curve in P^n, n >= 3 X: Ideal the ideal of a hypersurface containing the curve Outputs M: Module the H^1 of the normal sheaf of the curve with respect to the hypersurface Description Example S = ZZ/32009[x_0..x_3]; I = monomialCurveIdeal(S,{1,2,3}); X = ideal(gens I * random(source gens I, S^{1:-min(degrees source gens I)})); time h1NormalBundle(I,X) Caveat This function might require quite an amount of time to run, depending on the input data. /// doc /// Key alexanderSurface (alexanderSurface, Ring) Headline construct an Alexander surface Usage alexanderSurface S Inputs S: Ring the coordinate ring of P^4 Outputs : Sequence a pair (Y, pts) where Y is the ideal of an Alexander surface in P^4, i.e., the blow-up of P^2 along 10 points embedded in P^4 via |13L-4(p_1+...+p_{10})|, and pts is the list of the 10 blown-up points in P^2, saved for future use Description Example S = ZZ/32009[x_0..x_4]; time (Y, pts) = alexanderSurface S; betti res Y SeeAlso linSysAlexander properTransformAlexander /// doc /// Key linSysAlexander (linSysAlexander, ZZ, ZZ) Headline linear systems giving rise to curves of genus >= 16 on the Alexander surface Usage linSysAlexander(g,d) Inputs g: ZZ the genus of the desired curve d: ZZ the degree of the desired curve Outputs L: List Description Text If (g,d) belongs to ((16,17),(17,18),(18,19),(19,20),(20,20)), then the function returns a list (a,b_1,...,b_{10}) with the following feature: if we consider an Alexander surface, i.e., the blow up of P^2 along 10 general points p_i embedded in P^4 via |13L-4(p_1+...+p_{10})|, then the strict transform of a general curve in the linear system |a*L-b_1*p_1-...-b_{10}*p_{10}| will be a smooth curve of genus g and degree d in P^4 Example linSysAlexander(16,17) try linSysAlexander(12,14) else "not implemented" SeeAlso alexanderSurface properTransformAlexander /// doc /// Key properTransformAlexander (properTransformAlexander, Ring, List, List) Headline proper transform on the Alexander surface Usage properTransformAlexander(S, L, pts) Inputs S: Ring the coordinate ring of P^4 L: List a list (a,b_1,...,b_{10}) of 11 integers P: List a list of the ideals of 10 (general) points p_i in P^2 Outputs I: Ideal Description Text The function constructs a curve which is the proper transform of a general curve in P^2 in the linear system |a*L-b_1*p_1-...-b_{10}*p_{10}| via the blow-up map |13L-4(p_1+...+p_{10})| SeeAlso alexanderSurface linSysAlexander /// {* Example (g,d)=(16,17) L=linSysAlexander(g,d) S = ZZ/32009[x_0..x_4]; (Y,pts):=alexanderSurface(S); I=properTransformAlexander(S,L,pts); codim I, genus I, degree I *} doc /// Key auxiliarLineBundle (auxiliarLineBundle, Ideal, Ideal) Headline restriction of Omega(1) to a curve Usage auxiliarLineBundle(A,I) Inputs A: Ideal the ideal of a surface in P^4 I: Ideal the ideal of a curve on the surface Outputs : Module Description Text The function constructs the module representing the restriction to the curve of Omega(1), the twisted canonical bundle of the surface A /// doc /// Key singularCurveInP4 (singularCurveInP4, Ring, ZZ, ZZ) Headline nodal curve in P^4 of genus g and degree d Usage singularCurveInP4(S,g,d) Inputs S: Ring the coordinate ring of P^4 g: ZZ d: ZZ Outputs I: Ideal the ideal of a nodal curve in P^4 of genus g and degree d Description Text The function constructs a nodal curve of genus g and degree k:=2g-2-d in P^2 with a D=g^1_k and embeds it in P^4 via the complete linear system |K-D|. The construction of the nodal curve relies on the package @ HREF{"https://www.math.uni-sb.de/ag/schreyer/images/data/computeralgebra/kGonalNodalCurves.m2", "kGonalNodalCurves.m2"} @ by Christian Bopp SeeAlso curveGenus12Degree14InP4 curveGenus13Degree15InP4 randomCurveGenus12Degree14InP4 /// doc /// Key curveGenus12Degree14InP4 (curveGenus12Degree14InP4, Ring) Headline curve in P^4 of genus 12 and degree 14 Usage curveGenus12Degree14InP4 S Inputs S: Ring the coordinate ring of P^4 Outputs I: Ideal the ideal of a curve in P^4 of genus 12 and degree 14 Description Text The function constructs a nodal curve of genus 12 and degree 14 in P^4, then induces a matrix factorization on a random supporting cubic hypersurface and uses it to construct a new curve, which is in general smooth SeeAlso singularCurveInP4 curveGenus13Degree15InP4 /// doc /// Key curveGenus13Degree15InP4 (curveGenus13Degree15InP4, Ring) Headline curve in P^4 of genus 13 and degree 15 Usage curveGenus13Degree15InP4 S Inputs S: Ring the coordinate ring of P^4 Outputs I: Ideal the ideal of a curve in P^4 of genus 13 and degree 15 Description Text The function constructs a nodal curve of genus 13 and degree 15 in P^4, then induces a matrix factorization on a random supporting cubic hypersurface and uses it to construct a new curve, which is in general smooth SeeAlso singularCurveInP4 curveGenus12Degree14InP4 /// doc /// Key randomCurveGenus10Degree13InP4 (randomCurveGenus10Degree13InP4, Ring) Headline general curve in P^4 of genus 10 and degree 13 Usage randomCurveGenus10Degree13InP4 S Inputs S: Ring the coordinate ring of P^4 Outputs I: Ideal the ideal of a general curve in P^4 of genus 10 and degree 13 Description Text The function constructs a curve E of genus 10 on P^1 x P^2 via liaison techniques together with P={5 points on E}, then embeds E in P^4 via the complete linear system |K-P|. This construction is proven to be unirational and gives hence rise to a general curve of genus 10 and degree 13 /// doc /// Key randomCurveGenus12Degree14InP4 (randomCurveGenus12Degree14InP4, Ring) Headline general curve in P^4 of genus 12 and degree 14 Usage randomCurveGenus12Degree14InP4 S Inputs S: Ring the coordinate ring of P^4 Outputs I: Ideal the ideal of a general curve in P^4 of genus 12 and degree 14 Description Text The function constructs a general curve E of genus 10 and degree 13 in P^4 with @ TO{"randomCurveGenus10Degree13InP4"} @ and induces a matrix factorization on a random supporting cubic hypersurface, which is then used to construct a curve of genus 12 and degree 14. This construction is proven to be unirational SeeAlso singularCurveInP4 curveGenus12Degree14InP4 randomCurveGenus10Degree13InP4 /// doc /// Key randomGenus12Degree8CoverOfP1 (randomGenus12Degree8CoverOfP1, Ring) Headline general canonical curve of genus 12 with a g^1_8 Usage randomGenus12Degree8CoverOfP1 R Inputs S: Ring the coordinate ring of P^{11} Outputs : Sequence a triple (I,H_1,H_2) where I is the ideal of a random canonical curve of genus 12 and H_1 and H_2 are hyperplanes in P^{11} passing through 14 points on the curve. A general linear combination of H_1 and H_2 cuts the curve in these 14 points plus 8 points representing the g^1_8. Description Text The function constructs a general curve C of genus 12 and degree 14 in P^4 with @ TO{"randomCurveGenus12Degree14InP4"} @ and constructs its canonical embedding in P^{11}. By construction, if H is an hyperplane divisor in P^4, then K_C - H is a g^1_8. The function returns a sequence (I, H_1, H_2), where I is the ideal of the canonically embedded curve in P^{11} and H_1 and H_2 are the hyperplanes passing through 14 points on the curve corresponding to H via the canonical embedding. A general linear combination of H_1 and H_2 cuts the curve in these 14 points plus 8 points corresponding to the g^1_8. By construction, this procedure is unirational. SeeAlso randomCurveGenus12Degree14InP4 /// doc /// Key curveOnAlexanderSurface (curveOnAlexanderSurface, Ring, ZZ, ZZ) Headline curves on the Alexander surface Usage curveOnAlexanderSurface(S, g, d) Inputs S: Ring the coordinate ring of P^4 g: ZZ d: ZZ Outputs : Sequence a pair (I,Y) where I is the ideal of a curve of genus g and degree d and Y is an Alexander surface which contains it Description Text The function constructs a curve C of genus g and degree d for (g,d) belonging to ((16,17),(17,18),(18,19),(19,20),(20,20)). C is realized on the Alexander surface Y as the strict transform of a general element of a suitable linear system in P^2 SeeAlso alexanderSurface linSysAlexander properTransformAlexander /// doc /// Key isInBoijSoederbergCone (isInBoijSoederbergCone, BettiTally) Headline check whether a Betti table is in the Boij-Soederberg cone Usage isInBoijSoederbergCone B Inputs B: BettiTally Outputs : Boolean Description Text The function returns true if there exists a rational number q such that q*B is the Betti table of some module over a polynomial ring, and false if not Example B=new BettiTally from {(0, {1}, 1) => 6, (1, {2}, 2) => 10, (2, {3}, 3) => 3, (1, {3}, 3) => 3, (1, {4}, 4) => 1, (2, {5}, 5) => 13, (3, {6}, 6) => 9, (4, {7}, 7) => 1 } isInBoijSoederbergCone B B'=new BettiTally from {(0, {1}, 1) => 6, (1, {2}, 2) => 10, (2, {3}, 3) => 3, (1, {3}, 3) => 3, (1, {4}, 4) => 2, (2, {5}, 5) => 13, (3, {6}, 6) => 9, (4, {7}, 7) => 1 } isInBoijSoederbergCone B' /// doc /// Key listOfPossibleMatFac (listOfPossibleMatFac, BettiTally, ZZ) Headline different shapes of the same matrix factorization Usage listOfPossibleMatFac(B,d) Inputs B: BettiTally d: ZZ Outputs : List Description Text If (phi,psi) is a matrix factorization, then also (dual psi, dual phi), (psi, syz psi), and (dual phi, syz dual phi) are. This function provides a list of the shapes of the above matrix factorizations, including the original one Example p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; beta=betti map(S^{9:0,1:-1},S^{1:-1,9:-2},0) listOfPossibleMatFac(beta,3) listOfPossibleMatFac(beta,4) beta=betti map(S^{9:0,1:-1},S^{0:-1,10:-2},0) listOfPossibleMatFac(beta,3) /// doc /// Key candidateTables (candidateTables, BettiTally, ZZ) Headline Betti candidates for a given shape of a matrix factorization Usage bettiCandidates(B,d) Inputs B: BettiTally d: ZZ Outputs L: List Description Text The function returns a list of possible Betti tables, lying in the Boij-Soederberg cone and whose support has codimension >= 3, with the additional assumption that their projective dimension is at most 4. A module having a Betti table belonging to this list leads to a matrix factorization (phi,psi), if no cancellation occurs, such that one among the matrix factorizations (dual psi, dual phi), (psi, syz psi), (dual phi, syz dual phi), or (phi, psi) itself, has shape B Example p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; beta=betti map(S^{9:0,1:-1},S^{1:-1,9:-2},0) d=3; time candidateTables(beta,d) SeeAlso candidateTablesWithIrrelevantIdealAssociated /// doc /// Key candidateTablesWithIrrelevantIdealAssociated (candidateTablesWithIrrelevantIdealAssociated, BettiTally, ZZ) Headline Betti candidates with pd=5 for a given shape of a matrix factorization Usage bettiCandidatesIrrelevant(B,d) Inputs B: BettiTally d: ZZ Outputs : List Description Text The function returns a list of possible Betti tables, lying in the Boij-Soederberg cone and whose support has codimension >= 3, with the additional assumption that their projective dimension is 5. A module having a Betti table belonging to this list leads to a matrix factorization (phi,psi), if no cancellation occurs, such that one among the matrix factorizations (dual psi, dual phi), (psi, syz psi), (dual phi, syz dual phi), or (phi, psi) itself, has shape B Example p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; beta=betti map(S^{9:0,1:-1},S^{1:-1,9:-2},0) d=3; time candidateTablesWithIrrelevantIdealAssociated(beta,d) SeeAlso candidateTables /// doc /// Key arisingMatFac (arisingMatFac, BettiTally, ZZ) Headline shape of the arising matrix factorization assuming all possible cancellations Usage arisingMatFac(B,d) Inputs B: BettiTally d: ZZ Outputs : List Description Text The function produces the shape of the matrix factorization induced by a module with Betti table B on a hypersurface of degree d, assuming that all the possible cancellations occur Example p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; beta=betti map(S^{12:0,1:-1},S^{1:-1,12:-2},0) d=3; L=candidateTables(beta,3) apply(L,l->arisingMatFac(l,d)) /// doc /// Key precompiledListOfCandidates (precompiledListOfCandidates, ZZ) Headline list of Betti candidates for curve of genus g >= 16 Usage precompiledListOfCandidates g Inputs g: ZZ Outputs : List Description Text The function prints a precompiled list of the Betti candidates whose induced matrix factorizations have the same shape as the matrix factorization induced by a general curve of the input genus g and the degree as in the paper. The shape is computed assuming that every possible cancellation occurs Example p=32009; Fp=ZZ/p; S=Fp[x_0..x_4]; beta=betti map(S^{22:0,6:-2},S^{0:-1,4:-2,24:-3},0) possMF=listOfPossibleMatFac(beta,4); -- time pcand20=candidateTables(beta,4); -- takes about 53 seconds -- cand20=select(pcand20,B->any(possMF,l->l==arisingMatFac(B,4))); -- cand20 == precompiledListOfCandidates(20) precompiledListOfCandidates(20) /// doc /// Key verifyAssertionsOfThePaper (verifyAssertionsOfThePaper, ZZ) Headline print commands to verify the assertions of the paper Usage verifyAssertionsOfThePaper Description Text The function prints all the commands which are useful to check the assertions made in the paper. If the input is 0, all the commands are printed. Otherwise, depending on the input n between 1 and 8, the function prints the commands concerning particular assertions in the paper, as listed below: 0: all the assertions @ HREF("https://www.math.uni-sb.de/ag/schreyer/home/M2/doc/Macaulay2/MatFacCurvesP4/html/buffer01.html", "1: Proposition 2.1") @ @ HREF("https://www.math.uni-sb.de/ag/schreyer/home/M2/doc/Macaulay2/MatFacCurvesP4/html/buffer02.html", "2: Theorem 3.1, Remark 3.4, Corollary 3.5") @ @ HREF("https://www.math.uni-sb.de/ag/schreyer/home/M2/doc/Macaulay2/MatFacCurvesP4/html/buffer03.html", "3: Example 4.1") @ @ HREF("https://www.math.uni-sb.de/ag/schreyer/home/M2/doc/Macaulay2/MatFacCurvesP4/html/buffer04.html", "4: Proposition 4.2") @ @ HREF("https://www.math.uni-sb.de/ag/schreyer/home/M2/doc/Macaulay2/MatFacCurvesP4/html/buffer05.html", "5: Proposition 4.4") @ @ HREF("https://www.math.uni-sb.de/ag/schreyer/home/M2/doc/Macaulay2/MatFacCurvesP4/html/buffer06.html", "6: Theorem 5.1, Section 5.2") @ @ HREF("https://www.math.uni-sb.de/ag/schreyer/home/M2/doc/Macaulay2/MatFacCurvesP4/html/buffer07.html", "7: Table 5.2") @ @ HREF("https://www.math.uni-sb.de/ag/schreyer/home/M2/doc/Macaulay2/MatFacCurvesP4/html/buffer08.html", "8: Theorem 5.2") @ Every entry above is a hyperlink which leads to a file showing the execution on Macaulay2 of the corresponding commands, which can be used to preview the results of the computations and see what the output should look like. Warning: the code altogether might require some time to run. For instance, on our machine it amounted to a little more than 5 hours Example verifyAssertionsOfThePaper(0) /// end; restart uninstallPackage("MatFacCurvesP4") --uninstallPackage("kGonalNodalCurves") --installPackage("kGonalNodalCurves", RunExamples => false) installPackage("MatFacCurvesP4", RunExamples => false) installPackage("MatFacCurvesP4") --needsPackage("MatFacCurvesP4") viewHelp("MatFacCurvesP4") verifyAssertionsOfThePaper(0) --- loadPackage("MatFacCurvesP4")