Some special combinatorial classes are predefined in the combstruct package. They are Combination or Subset, Permutation, Partition, and Composition.

Combination or Subset:

Combinations (or subsets) of elements

The argument is a set, list, or non-negative integer.  In the case of a non-negative integer n, it is treated as the set $\{1,2,...,n\}$.  The default size is $'\mathrm{allsizes}'$.

Use the string 'allsizes' when the object is selected from all possible sizes. If the size is not specified, each structure has default behavior.

Permutation:

Permutations of elements

The argument is a set, list, or non-negative integer.  In the case of a non-negative integer n, it is treated as the list $\[1,2,...,n\]$.  The default size is the number of elements in the set/list.

Partition:

Partitions of positive integers into sums of positive integers without regards to order.

The argument is a positive integer.  The default size is 'allsizes'.

Use the string 'allsizes' when the object is selected from all possible sizes. If the size is not specified, each structure has default behavior.

Composition:

Composition of a positive integer n into a positive integer k parts of size at least 1, where the order of summands is meaningful.

The argument is a positive integer.  The default size is $'\mathrm{allsizes}'$.

Use the string 'allsizes' when the object is selected from all possible sizes. If the size is not specified, each structure has default behavior.

Combination and Subset are different names for the same structure.

You can define your own structure that is understood by the functions in the combstruct package. To create the structure Foo, create the procedures `combstruct/count/Foo`, `combstruct/draw/Foo`, `combstruct/allstructs/Foo`, and `combstruct/iterstructs/Foo`.

Each of these functions must take size as their first argument, a non-negative integer, the string $'\mathrm{allsizes}'$ or the string $'\mathrm{default}'$, with the arguments that your structure needs. Thus, $\mathrm{draw}\left(\mathrm{Foo}\left(x\,y\,z\right)\,\mathrm{size}=n\right)$ invokes `combstruct/draw/Foo`(n, x, y, z).

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

$\mathrm{count}\left(\mathrm{Subset}\left(\left\{1\,2\,3\right\}\right)\right)$

$8$

$\mathrm{draw}\left(\mathrm{Combination}\left(5\right)\,\mathrm{size}=4\right)$

$\left\{1\,2\,3\,4\right\}$

$\mathrm{count}\left(\mathrm{Permutation}\left(\left[a\,a\,b\right]\right)\right)$

$3$

$\mathrm{it}≔\mathrm{iterstructs}\left(\mathrm{Permutation}\left(\left[a\,a\,b\right]\right)\,\mathrm{size}=2\right)\:$

$\mathrm{draw}\left(\mathrm{Partition}\left(9\right)\right)$

$\left[2\,2\,5\right]$

$\mathrm{allstructs}\left(\mathrm{Composition}\left(3\right)\,\mathrm{size}=2\right)$

$\left[\left[1\,2\right]\,\left[2\,1\right]\right]$