On Combinatorial Structures Kept Fixed by the Action of a Given Permutation

Let β be a given permutation of n]={1,2,...,n} of type (β₁, β₂,...,β_n) (i.e., β has β₁ cycles of length i; ∑iβ₁ = n. We find (in terms of the β₁'s and bijectively) the number of endofunctions, permutations, cyclic permutations, derangements, fixed point free involutions, assemblies of octopuses, octopuses, idempotent endofunctions, rooted trees (i.e. contractions), forests of rooted trees, trees, vertebrates, relations (digraphs), symmetric relations (simple graphs), partitions, and connected endofunctions on n], kept fixed by the natural action (byconjugation) of β. This approach leads to algorithms generating these structures.