PartitionQ[p] yields True if p is an integer partition. PartitionQ[n, p] yields True if p is a partition of n.

PermutationWithCycle[n, {i, j, ...}] gives a size-n permutation in which {i, j, ...} is a cycle and all other elements are fixed points.

RandomKSetPartition[set, k] returns a random set partition of set with k blocks. RandomKSetPartition[n, k] returns a random set partition of the first n natural numbers into ...

RandomSetPartition[set] returns a random set partition of set. RandomSetPartition[n] returns a random set partition of the first n natural numbers.

SetPartitions[set] returns the list of set partitions of set. SetPartitions[n] returns the list of set partitions of {1, 2, ..., n}.

Turan[n, p] constructs the Turán graph, the extremal graph on n vertices that does not contain CompleteGraph[p].

UnrankBinarySubset[n, l] gives the n\[Null]^th subset of list l, listed in increasing order of integers corresponding to the binary representations of the subsets.

UnrankSetPartition[r, s, k] finds a k-block set partition of s with rank r. UnrankSetPartition[r, n, k] finds a k-block set partition of {1, 2, ..., n} with rank r.

UnrankPermutation[r, l] gives the r\[Null]^th permutation in the lexicographic list of permutations of list l. UnrankPermutation[r, n] gives the r\[Null]^th permutation in ...

UnrankSetPartition[r, set] finds a set partition of set with rank r. UnrankSetPartition[r, n] finds a set partition of {1, 2, ..., n} with rank r.