NumberOfInvolutions[n] counts the number of involutions on n elements.

NumberOfPartitions[n] counts the number of integer partitions of n.

NumberOfPermutationsByCycles[n, m] gives the number of permutations of length n with exactly m cycles.

NumberOfPermutationsByTypes[l] gives the number of permutations of type l.

NumberOfSpanningTrees[g] gives the number of labeled spanning trees of graph g.

PermutationGraph[p] gives the permutation graph for the permutation p.

PermutationGroupQ[l] yields True if the list of permutations l forms a permutation group.

PermutationQ[p] yields True if p is a list representing a permutation and False otherwise.

PermutationToTableaux[p] returns the tableaux pair that can be constructed from p using the Robinson\[Dash]Schensted\[Dash]Knuth correspondence.

PermutationType[p] returns the type of permutation p.