Indecomposable Permutations, Hypermaps and Labeled Dyck Paths

TL;DR

This paper establishes bijections between hypermaps, indecomposable permutations, and labeled Dyck paths, leading to new enumerative formulas for hypermaps and insights into permutation parameters.

Contribution

It introduces a bijection between indecomposable permutations and labeled Dyck paths, enabling new enumeration formulas for hypermaps and related combinatorial objects.

Findings

Derived an inductive formula for counting hypermaps with given parameters.

Connected hypermaps to indecomposable permutations and Dyck paths.

Analyzed the distribution of permutation parameters such as cycles and maxima.

Abstract

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.