NumberOfPermutationsByInversions[n, k] gives the number of permutations of length n with exactly k inversions. NumberOfPermutationsByInversions[n] gives a table of the number ...

OutDegree[g, n] returns the out-degree of vertex n in directed graph g. OutDegree[g] returns the sequence of out-degrees of the vertices in directed graph g.

PairGroup[g] returns the group induced on 2-sets by the permutation group g. PairGroup[g, Ordered] returns the group induced on ordered pairs with distinct elements by the ...

PartialOrderQ[g] yields True if the binary relation defined by edges of the graph g is a partial order, meaning it is transitive, reflexive, and antisymmetric. ...

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

PerfectQ[g] yields True if g is a perfect graph, meaning that for every induced subgraph of g the size of a largest clique equals the chromatic number.

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

PetersenGraph returns the Petersen graph, a graph whose vertices can be viewed as the size-2 subsets of a size-5 set with edges connecting disjoint subsets.

RadialEmbedding[g, v] constructs a radial embedding of the graph g in which vertices are placed on concentric circles around v depending on their distance from v. ...

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 ...