A new combinatorial identity for unicellular maps, via a direct bijective approach

TL;DR

This paper introduces a new combinatorial identity for counting unicellular maps using a bijective approach, simplifying the recursive structure and providing explicit formulas and bijections to well-understood combinatorial objects.

Contribution

It presents a novel recursive decomposition of unicellular maps into smaller genus maps, leading to an explicit bijection with plane trees and simplified counting formulas.

Findings

Derived a new recursive identity for $psilon_g(n)$

Established a bijection between unicellular maps and plane trees with distinguished vertices

Provided a simple formula for the polynomial factor in the count of unicellular maps

Abstract

A unicellular map, or one-face map, is a graph embedded in an orientable surface such that its complement is a topological disk. In this paper, we give a new viewpoint to the structure of these objects, by describing a decomposition of any unicellular map into a unicellular map of smaller genus. This gives a new combinatorial identity for the number $\epsilon_g(n)$ of unicellular maps of size $n$ and genus $g$. Contrarily to the Harer-Zagier recurrence formula, this identity is recursive in only one parameter (the genus). Iterating the construction gives an explicit bijection between unicellular maps and plane trees with distinguished vertices, which gives a combinatorial explanation (and proof) of the fact that $\epsilon_g(n)$ is the product of the $n$-th Catalan number by a polynomial in $n$. The combinatorial interpretation also gives a new and simple formula for this polynomial. Variants of the problem are considered, like bipartite unicellular maps, or unicellular maps with cubic vertices only.