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

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Let β be a given permutation of [n]={1,2,...,n} of type (β 1, β 2,...,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.
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Document Type: Research Article

Affiliations: Université du Québec à Montréal

Publication date: February 1, 1991

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