Abstract
This paper addresses the enumeration of rooted and unrooted hypermaps of a given genus. For rooted hypermaps the enumeration method consists of considering the more general family of multirooted hypermaps, in which darts other than the root dart are distinguished. We give functional equations for the generating series counting multirooted hypermaps of a given genus by number of darts, vertices, edges, faces and the degrees of the vertices containing the distinguished darts. We solve these equations to get parametric expressions of the generating functions of rooted hypermaps of low genus. We also count unrooted hypermaps of given genus by number of darts, vertices, hyperedges and faces.
Highlights
A hypermap is a triple (D, R, L) where D is a finite set of darts and R and L are permutations on D such that the group R, L generated by R and L acts transitively on D
It is of genus 0, has 1 face, its one vertex is of degree 0 and its list D is empty because it has no distinguished vertices; so it constitutes the singleton
Since the root of an incidence map of a rooted hypermap must belong to a white vertex, we impose the condition on a rooted 2-face-coloured face-bipartite map that the root belong to a white face and we transform (2.2) – (2.6) into the corresponding bijective decomposition for these maps
Summary
A (combinatorial) hypermap is a triple (D, R, L) where D is a finite set of darts and R and L are permutations on D such that the group R, L generated by R and L acts transitively on D. We state (in Section 4) a bijective decomposition for the set H(g, t, f, e, n, D) of sequenced orientable hypermaps of genus g with t darts, f faces and e hyperedges, with the root vertex of degree n and with the sequence of degrees of the distinguished vertices equal to D = We obtain a bijective decomposition of the set F(g, e, w, b, n, D) of sequenced orientable face-bipartite maps of genus g with e edges, w white faces, b black faces, with the root face of degree 2n and with the sequence of half-degrees of the distinguished vertices equal to D. We relate multirooted hypermaps to sequenced hypermaps and obtain a recurrence for the number of multirooted hypermaps and functional equations for the generating series counting multirooted hypermaps of a given genus by number of darts, vertices, edges, faces and the degrees of the initial vertices of the distinguished darts. B) contains a table for numbers of rooted (resp. unrooted) hypermaps of genus g with d darts, v vertices and e hyperedges for d ≤ 14
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