The setpartition command returns all the partitions of S. A partition of a set S is a set of subsets of S such that every element s in S is in one and only one of these subsets of the partition, and the union of these subsets is equal to S. A set of $n$ elements has combinat[bell](n) partitions (the $n$th Bell number).

If S is a list of objects, then each partition in the output is represented as a list of sublists instead of a set of subsets. By passing the optional argument returnsets = true, each partition in the output will be a set of subsets even if S is a list. Likewise, when passing returnsets = false, each partition will be a list of sublists even if S is a set.

Note: If S is a list of $n$ elements, with some of them being repeated, the total number of partitions will be smaller than combinat[bell](n) and some elements s of S will appear in more than one sublist of a single partition. For example, when computing all the partitions of $\left[1\,1\,2\right]$ via setpartition([1, 1, 2]), the partition $\left[\left[1\right]\,\left[1\,2\right]\right]$ appears only once in the result and it contains the element $1$ in two sublists.

When the optional argument m is given, each subset is of size m where m must divide the cardinality of the set.

$\mathrm{with}\left(\mathrm{combinat}\,\mathrm{setpartition}\right)$

$\left[\mathrm{setpartition}\right]$

$S≔\left\{1\,2\,3\,4\right\}$

$S≔\left\{1\,2\,3\,4\right\}$

$\mathrm{setpartition}\left(S\right)$

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

$\mathrm{nops}\left(\right)=\mathrm{combinat}:-\mathrm{bell}\left(4\right)$

$15=15$

$\mathrm{nops}\left(\mathrm{setpartition}\left(\left[1\,2\,3\,4\,5\,6\,7\right]\right)\right)=\mathrm{combinat}:-\mathrm{bell}\left(7\right)$

$877=877$

$\mathrm{setpartition}\left(S\,2\right)$

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

$S≔\left[1\,1\,3\,4\right]$

$S≔\left[1\,1\,3\,4\right]$

$\mathrm{setpartition}\left(S\,2\right)$

$\left[\left[\left[1\,1\right]\,\left[3\,4\right]\right]\,\left[\left[1\,3\right]\,\left[1\,4\right]\right]\right]$

The combinat[setpartition] command was updated in Maple 18.

The returnsets option was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.