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The editing window is hopefully enough self-descriptive. For more information about dokuwiki syntax, see Formatting Syntax. In addition, please pay attention to the following two features: citations and formulas.

For rendering formulas, we use a modification of the latex plugin for dokuwiki. Inline formulas are to be written inside single dollar signs. Formulas on separate line are to be written inside double dollar signs.

Each formula is compiled as a separate LaTeX document and the result is then transformed to a SVG image. Note that it is not possible to define any additional commands locally for each page.

We define the following additional macros. The purpose of those definitions is not only to make writing formulas easier but also to unify the notation for the wiki. So, authors are strongly encouraged to use the macros and to use them in the meaning specified in the table. If you suggest any changes in the list, please contact the administrators.

Symbol	Command	Meaning
	\C	Set of complex numbers
	\Cscr	Some category
	\FinHilb	Category of finite-dimensional Hilbert spaces
	\FinVect	Category of finite-dimensional vector spaces
	\FundRep	Category generated by the fundamental representation
	\GL	General linear group
	\Hilb	Category of Hilbert spaces
	\Kscr	Some category
	\N	Set of natural numbers (without zero)
	\Pscr	Set of all partitions
	\Part	Linear category of all partitions
	\R	Set of real numbers
	\Rep	Category of representations
	\spanlin	Linear span
	\Q	Set of rational numbers
	\Vect	Category of vector spaces
	\Z	Set of integers

For typesetting partitions, we use the macro package partmac from Daniel Gromada. Here are the instructions

There are macros of the form \Lxxxx for partitions on lower line, \Uxxxx for partitions on upper line and \Pxxxx for partitions with the same ammount of upper and lower points. Here xxxx is the lexicographically smallest word representation of the partiiton. That is, we have the following macros for partitions with points on the lower line.

\La		\Laaaa		\Laabc
\Laa		\Laaab		\Labac
\Lab		\Laaba		\Labca
\Laaa		\Labaa		\Labbc
\Laab		\Labbb		\Labcb
\Laba		\Laabb		\Labcc
\Labb		\Labba		\Labcd
\Labc		\Labab

Then we have the following macros for partitions with points on the upper line.

\Ua		\Uaaaa		\Uaabc
\Uaa		\Uaaab		\Uabac
\Uab		\Uaaba		\Uabca
\Uaaa		\Uabaa		\Uabbc
\Uaab		\Uabbb		\Uabcb
\Uaba		\Uaabb		\Uabcc
\Uabb		\Uabba		\Uabcd
\Uabc		\Uabab

Finally partitions with equal number of points on lower and upper line.

\Paa		\Pabab		\Pabcabc
\Pab		\Paabc		\Pabcabd
\Paaaa		\Pabac		\Pabcadc
\Paaab		\Pabca		\Pabcdbc
\Paaba		\Pabbc		\Pabcade
\Pabaa		\Pabcb		\Pabcdbe
\Pabbb		\Pabcc		\Pabcdec
\Paabb		\Pabcd		\Pabcdef
\Pabba				\Paabaab

In addition, we define the following synonyms.

\singleton		\idpart		\fourpart
\upsingleton		\disconnecterpart		\crosspart
\pairpart		\positionerpart		\halflibpart
\uppairpart		\connecterpart

To define a general partition with points only on the lower or upper line, one can use macro \Lpartition, resp. \Upartition. The syntax is the following.

\LPartition{<singletons>}{<remaining blocks>}

The datum <singletons> should be of the form , where is the height of the singleton blocks and are the positions of the singletons. The datum <remaining blocks> consists of descriptions of other blocks. Each block is described similarly as the set of singletons, so in the format , where is the height of the block and are the positions of elements of the block. The data for the blocks are separated by semicolon.

Let us show this on example.

\LPartition{0.4:1,4,8}{0.4:2,3;0.4:5,7;0.8:6,9,10}

Here, the singletons are on position 1, 4, 8 and each of them is represented by a line of height 0.4. Then there are three additional blocks. First connecting points 2 and 3 is represented by a node of height 0.4 (that is, the same as the singletons). Second block connects points 5 and 7 and has the same height. Finally a block connecting points 6, 9, 10 has double height, that is, 0.8.

The macro \UPartition works the same. Except that instead of the height, we should put , that is, we put there actually the -coordinate of the point. That means, to obtain the same result horizontally flipped, we have to write down

\UPartition{0.6:1,4,8}{0.6:2,3;0.6:5,7;0.2:6,9,10}

The units are chosen in such a way that one should keep the height between 0 and 1 to stick within the line in a paragraph. However, the macro works also if you put there higher numbers, which can be used especially in display mode. For example

\LPartition{0.6:1,4,8}{0.6:2,3;0.6:5,7;1.2:6,9,10}

To draw general partitions with upper and lower points, one can use \Partition{<data>}. The data can consist of the following commands.

\Psingletons y1 to y2:i1,i2,...,ik    %draws singletons
\Pblock      y1 to y2:i1,i2,...,ik    %draws one block
\Pline       (x1,y1) (x2,y2)          %draws a line

Here, x1 and x2 represent the coordinates (i.e. position of a point) and y1 and y2 the -coordinates. Again, one is advised to keep the coordinates between 0 and 1. As an example, we mention the definition of the connecter partition and the positioner partition .

\Partition{                 % connecter partition
\Pblock 0 to 0.3:1,2        % connecting two lower points
\Pblock 1 to 0.7:1,2        % connecting two upper points
\Pline (1.5,0.3) (1.5,0.7)  % connecting the two blocks together
}

\Partition{               % positioner partition
\Psingletons 0to0.3:2     % singleton on lower line, pos. 2
\Psingletons 1to0.7:1     % singleton on upper line, pos. 1
\Pline (1,0) (2,1)        % line connecting lower pt 1 and upper pt 2
}

For drawing more complicated partitions, one can use the \BigPartition{<data>}, which works exactly the same, but produces a larger result. Another difference is that \BigPartition aligns the middle of the partition, i.e. the point with the equals sign. So, for example the result

can be obtained writing

$$p=
\BigPartition{
\Pblock 0 to 0.25:2,3
\Pblock 1 to 0.75:1,2,3
\Psingletons 0 to 0.25:1,4
\Pline (2.5,0.25) (2.5,0.75)
},
\qquad
q=
\BigPartition{
\Psingletons 0 to 0.25:1,4
\Psingletons 1 to 0.75:1,4
\Pline (2,0) (3,1)
\Pline (3,0) (2,1)
\Pline (2.75,0.25) (4,0.25)
}$$

To add text, one can use \Ptext(<x>,<y>){<text>}, where '<x>' and '<y>' are coordinates and '<text>' is any code. The '<text>' is wrapped in a box, whose center is described by the coordinates. An example:

is obtained writing

$$p=
\BigPartition{
\Pblock 0 to 0.25:2,3
\Pblock 1 to 0.75:1,2,3
\Psingletons 0 to 0.25:1,4
\Pline (2.5,0.25) (2.5,0.75)
\Ptext(1,1.2){1}
\Ptext(2,1.2){2}
\Ptext(3,1.2){3}
\Ptext(1,-0.2){1}
\Ptext(2,-0.2){2}
\Ptext(3,-0.2){3}
\Ptext(4,-0.2){4}
}$$

In this section, we describe how to assign different shapes to the set of partitioned points to obtained so-called colored partitions. In the package, we prepared two colors. Command \Pw is used to draw white circle and \Pb is used to draw black circle .

For partitions with lower or upper points only, one can use \LPartition resp. \UPartition and specify the colorings in the <singletons> parameter. An example:

\LPartition{0.6:1,4,8;\Pw:1,2,5,6;\Pb:3,4,7,8,9,10}{0.6:2,3;0.6:5,7;1.2:6,9,10}

To add points inside \Partition or \Bigpartition, one can use command of the form \Ppoint <y> <shape>:<positions> as in the following example

\BigPartition{
\Psingletons 0 to 0.25:1,4
\Psingletons 1 to 0.75:1,4
\Pline (2,0) (3,1)
\Pline (3,0) (2,1)
\Pline (2.75,0.25) (4,0.25)
\Ppoint0 \Pw:2,4
\Ppoint0 \Pb:1,3
\Ppoint1 \Pw:1,2,3
\Ppoint1 \Pb:4
}

Actually, we have an additional two pre-defined points, which are actually arrows. That is, \Ls for arrow up (letters stand for lower singleton) and \Us for arrow down. Using them, we can draw the colored singleton by \LPartition{\Ls:1;\Pw:1}{}. If we wanted to emphasize the singletons in the example above, we can also replace them by arrows.