A generalization of the quadrangulation relation to constellations and hypermaps

Constellations and hypermaps generalize combinatorial maps, 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) stating an enumerative relation between quadrangulations and bipartite quadrangulations. We show a similar relation between hypermaps and constellations by using a result of Littlewood on factorization of characters. A combinatorial proof of Littlewood's result is also given. Furthermore, we show that coefficients in our relation are all positive integers, hinting possibility of a combinatorial interpretation. Using this enumerative relation, we recover a result on the asymptotic behavior of hypermaps in Chapuy (2009).