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

NumberOfTableaux[p] uses the hook length formula to count the number of Young tableaux with shape defined by partition p.

OddGraph[n] returns the graph whose vertices are the size-(n - 1) subsets of a size-(2 nTraditionalForm\`\[Dash]1) set and whose edges connect pairs of vertices that ...

One is a tag used in several functions to inform the functions that only one object need be considered or only one solution be produced, as opposed to all objects or all ...

OrientGraph[g] assigns a direction to each edge of a bridgeless, undirected graph g, so that the graph is strongly connected.

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 ...

ParentsToPaths[l, i, j] takes a list of parents l and returns the path from i to j encoded in the parent list. ParentsToPaths[l, i] returns the paths from i to all vertices.

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.