probOfFactor -- probability that a polynomial of degree n is square free AND has a factor of degree k over the same finite ground field

The probability that a monic polynomial of degree n over a finite field F of q elements is square free AND has a factor of degree d over F, can be computed via a over partitions. An approximation for q-> infinity is as the relative size of the conjugacy classes with a sub partitions of size k in the symmetric group S_n. The first version returns the approximation. The second version returns the precise value. The last version returns the probability as function of q up to order q ^-ord.

i1 : probOfFactor(30,10)

o1 = .385481092947806

o1 : RR (of precision 53)

i2 : probOfFactor(6,3)

o2 = .3625

o2 : RR (of precision 53)

i3 : probOfFactor(6,3,101)

o3 = .35572340337224

o3 : RR (of precision 53)

i4 : q=symbol q; Q = RR[q, MonomialOrder => Lex, Inverses=>true]

o5 = Q

o5 : PolynomialRing

i6 : probOfFactor(6,3,Q,1)

                   -1
o6 = .3625 - .6875q

o6 : Q

i7 : probOfFactor(12,1,Q,1)

                      -1
o7 = .632121 - .81606q

o7 : Q