Relating ordinary and fully simple maps via monotone Hurwitz numbers

Understand

A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors.

• The relation is via monotone Hurwitz numbers and was originally proved using Weingarten calculus for matrix integrals.
• The goal of this paper is to present two independent proofs that are purely combinatorial and generalise in various directions, such as to the setting of stuffed maps and hypermaps.
• The main motivation to understand the relation between ordinary and fully simple maps is the fact that it could shed light on fundamental, yet still not well-understood, problems in free probability and topological recursion.