Bijective enumeration of bicolored maps of given vertex degree distribution

We derive a new formula for the number of factorizations of a full cycle into an ordered product of two permutations of given cycle types. For the first time, a purely combinatorial argument involving a bijective description of bicolored maps of specified vertex degree distribution is used. All the previous results in the field rely either partially or totally on a character theoretic approach. The combinatorial proof relies on a new bijection extending the one in [G. Schaeffer and E. Vassilieva. J. Comb. Theory Ser. A, 115(6):903-924, 2008] that focused only on the number of cycles. As a salient ingredient, we introduce the notion of thorn trees of given vertex degree distribution which are recursive planar objects allowing simple description of maps of arbitrary genus.