Indecomposable Permutations, Hypermaps and Labeled Dyck Paths

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.Comment: 30 pages 4 Figures. submitte

Hypermaps and indecomposable permutations

AbstractIt is shown that the number of hypermaps of size n, that is the number of ordered pairs of permutations generating a transitive subgroup of Sn, is equal to (n−1)! times the number of indecomposable permutations of Sn+1. The proof is elementary

Some Combinatorial Aspects of Time-stamp Systems

AbstractThe aim of this paper is to outline a combinatorial structure appearing in distributed computing, namely a directed graph in which a certain family of subsets with k vertices have a successor. It has been proved that the number of vertices of such a graph is at least 2k - 1 and an effective construction has been given which needs k2k - 1 vertices. This problems is issued from some questions related to the labeling of processes in a system for determining the order in which they were created. By modifying some requirements on the distributed system, we show that there arise other combinatorial structures leading to the construction of solutions the size of which becomes a linear function of the input

A Whitney polynomial for hypermaps

We introduce a Whitney polynomial for hypermaps and use it to generalize the results connecting the circuit partition polynomial to the Martin polynomial and the results on several graph invariants

Odd permutations are nicer than even ones

International audienceWe give simple combinatorial proofs of some formulas for the number of factorizations of permutations in S n as a product of two n-cycles, or of an n-cycle and an (n−1)-cycle. ... The parameter number of cycles plays a central role in the algebraic theory of the symmetric group, however there are very few results giving a relationship between the number of cycles of two permutations and that of their product. ... The first results on the subject go back to Ore, Bertram, Stanley (see [13], [1] and [15]), who proved some existence theorems. These results allowed to obtain ..