# Relating ordinary and fully simple maps via monotone Hurwitz numbers

## TL;DR

This paper presents two combinatorial proofs linking ordinary and fully simple maps through monotone Hurwitz numbers, expanding understanding in map enumeration and its connections to free probability and topological recursion.

## Contribution

It provides new combinatorial proofs of the relation between ordinary and fully simple maps, generalizing to stuffed maps and hypermaps.

## Key findings

- Two independent combinatorial proofs established
- Generalization to stuffed maps and hypermaps achieved
- Enhanced understanding of map enumeration's role in free probability

## Abstract

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.