# Combinatorics of Bousquet-M\'elou-Schaeffer numbers in the light of topological recursion

## TL;DR

This paper proves a combinatorial quasi-polynomiality property for Bousquet-Mélou-Schaeffer numbers and relates it to topological recursion, extending the connection between combinatorics and mathematical physics.

## Contribution

It establishes a purely combinatorial proof of a structural property and generalizes topological recursion to higher order critical points for these numbers.

## Key findings

- Proves quasi-polynomiality of Bousquet-Mélou-Schaeffer numbers
- Derives topological recursion for higher order critical points
- Links combinatorial properties with spectral curve analysis

## Abstract

In this paper we prove, in a purely combinatorial way, a structural quasi-polynomiality property for the Bousquet-M\'elou-Schaeffer numbers. Conjecturally, this property should follow from the Chekhov-Eynard-Orantin topological recursion for these numbers (or, to be more precise, the Bouchard-Eynard version of the topological recursion for higher order critical points), which we derive in this paper from the recent result of Alexandrov-Chapuy-Eynard-Harnad. To this end, the missing ingredient is a generalization to the case of higher order critical points on the underlying spectral curve of the existing correspondence between the topological recursion and Givental's theory for cohomological field theories.