A generalization of the quadrangulation relation to constellations and hypermaps

Abstract

Constellations and hypermaps generalize combinatorial maps, $\textit{i.e.}$ embedding of graphs in a surface, in terms of factorization of permutations. In this paper, we extend a result of Jackson and Visentin (1990) on an enumerative relation between quadrangulations and bipartite quadrangulations. We show a similar relation between hypermaps and constellations by generalizing a result in the original paper on factorization of characters. Using this enumerative relation, we recover a result on the asymptotic behavior of hypermaps of Chapuy (2009).

Keywords

• combinatorial maps
• constellations
• enumeration
• character factorization
• [info.info-dm]computer science [cs]/discrete mathematics [cs.dm]