Indecomposable Permutations, Hypermaps and Labeled Dyck Paths

Details Indecomposable Permutations, Hypermaps and Labeled Dyck Paths Indecomposable Permutations, Hypermaps and Labeled Dyck Paths

2008-12-02

arxiv.org/abs/0812.0440v1

Mathematics

Combinatorics

30 pages 4 Figures. submitted

Scientific paper

Hypermaps were introduced as an algebraic tool for the representation of embeddings of graphs on an orientable surface. Recently a bijection was given between hypermaps and indecomposable permutations; this sheds new light on the subject by connecting a hypermap to a simpler object. In this paper, a bijection between indecomposable permutations and labelled Dyck paths is proposed, from which a few enumerative results concerning hypermaps and maps follow. We obtain for instance an inductive formula for the number of hypermaps with n darts, p vertices and q hyper-edges; the latter is also the number of indecomposable permutations of with p cycles and q left-to-right maxima. The distribution of these parameters among all permutations is also considered.

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