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A direct relation between the enumeration of ordinary maps and that of fullysimple maps first appeared in the work of the first and last authors. Therelation is via monotone Hurwitz numbers and was originally proved usingWeingarten calculus for matrix integrals. The goal of this paper is to presenttwo independent proofs that are purely combinatorial and generalise in variousdirections, such as to the setting of stuffed maps and hypermaps. The mainmotivation to understand the relation between ordinary and fully simple maps isthe fact that it could shed light on fundamental, yet still notwell-understood, problems in free probability and topological recursion.

Relating ordinary and fully simple maps via monotone Hurwitz numbers