A combinatorial class is either an elementary class, or is built from simpler classes with "constructors." The elementary classes are Epsilon, which represents an object of size zero, and Atom, which represents an object of size one. The available constructors are listed in the following table.

For the constructors Set, PowerSet, Sequence, and Cycle, it is possible to add restrictions on the cardinality. For example, $\mathrm{Set}\left(A\,1\le \mathrm{card}\right)$ means all nonempty sets whose elements are in $A$, $\mathrm{Sequence}\left(A\,\mathrm{card}\le 3\right)$ means all sequences of at most three elements of $A$, and $\mathrm{Cycle}\left(A\,\mathrm{card}=5\right)$ means all directed cycles of five elements from $A$.

None of these constructors accept an object that generates Epsilon as an argument. In some cases, no cardinality restriction or $k\le \mathrm{card}$ with any constructor but PowerSet, such a grammar generates an infinite number of objects of size 0, whereas this system is for classes with a finite number of members of each size.

In the others, PowerSet or $\mathrm{card}\le k$ or $\mathrm{card}=k$, while there are only a finite number of objects of size 0, the current system does not handle grammars with such Epsilon productions.

In $\mathrm{Subst}\left(A\,B\right)$, neither $A$ nor $B$ may produce objects of size 0.

A specification is a set of productions of the form $A=\mathrm{rhs}$, where $A$ is the name of the class being defined, and $\mathrm{rhs}$ is an expression involving elementary classes, constructors, and other classes. For example, the following table lists specifications of some well-known combinatorial classes.

When the labeling type is 'labeled'.

When the labeling type is 'unlabeled'.

It is possible to use Epsilon as a way of marking certain objects without affecting their size.

There are also predefined structures, for example, combinations, built into the system. For more information, see combstruct[structures].

$\mathrm{with}\left(\mathrm{combstruct}\right)\:$

$\mathrm{sys}≔\left\{C=\mathrm{Sequence}\left(\mathrm{Prod}\left(a\,C\,b\right)\right)\,a=\mathrm{Atom}\,b=\mathrm{Atom}\right\}\:$

$\mathrm{word}≔\mathrm{draw}\left(\left[C\,\mathrm{sys}\right]\,\mathrm{size}=6\right)$

$\mathrm{word}≔\mathrm{Sequence}\left(\mathrm{Prod}\left(a\,\mathrm{Sequence}\left(\mathrm{Prod}\left(a\,\mathrm{{\rm E}}\,b\right)\right)\,b\right)\,\mathrm{Prod}\left(a\,\mathrm{{\rm E}}\,b\right)\right)$

$\mathrm{eval}\left(\mathrm{subs}\left(\mathrm{`=`}\left(\mathrm{Prod}\,\left(\right)\mapsto \mathrm{args}\right)\,\mathrm{`=`}\left(\mathrm{Sequence}\,\left(\right)\mapsto \mathrm{args}\right)\,\mathrm{{\rm E}}=\mathrm{NULL}\,\mathrm{word}\right)\right)$

$a,a,b,b,a,b$

$\mathrm{circuit}≔\left\{C=\mathrm{Union}\left(P\,S\,R\right)\,P=\mathrm{Set}\left(\mathrm{Union}\left(S\,R\right)\,2\le \mathrm{card}\right)\,R=\mathrm{Atom}\,S=\mathrm{Set}\left(\mathrm{Union}\left(P\,R\right)\,2\le \mathrm{card}\right)\right\}\:$

$\mathrm{draw}\left(\left[C\,\mathrm{circuit}\,\mathrm{labeled}\right]\,\mathrm{size}=5\right)$

$\mathrm{Set}\left(R_4\,\mathrm{Set}\left(\mathrm{Set}\left(R_5\,\mathrm{Set}\left(R_1\,R_2\right)\right)\,R_3\right)\right)$

$\mathrm{circuit2}≔\left\{C=\mathrm{Union}\left(P\,S\,R\right)\,P=\mathrm{Prod}\left(\mathrm{par}\,\mathrm{Set}\left(\mathrm{Union}\left(S\,R\right)\,2\le \mathrm{card}\right)\right)\,R=\mathrm{Atom}\,S=\mathrm{Prod}\left(\mathrm{ser}\,\mathrm{Set}\left(\mathrm{Union}\left(P\,R\right)\,2\le \mathrm{card}\right)\right)\,\mathrm{par}=\mathrm{{\rm E}}\,\mathrm{ser}=\mathrm{{\rm E}}\right\}\:$

$\mathrm{draw}\left(\left[C\,\mathrm{circuit2}\,\mathrm{labeled}\right]\,\mathrm{size}=5\right)$

$\mathrm{Prod}\left(\mathrm{ser}\,\mathrm{Set}\left(R_2\,\mathrm{Prod}\left(\mathrm{par}\,\mathrm{Set}\left(R_3\,R_4\right)\right)\,\mathrm{Prod}\left(\mathrm{par}\,\mathrm{Set}\left(R_5\,R_1\right)\right)\right)\right)$