A simple model of trees for unicellular maps

TL;DR

This paper introduces a new bijection that simplifies the representation of unicellular maps as pairs of trees and permutations, providing unified and bijective proofs for various counting formulas and extending to related structures.

Contribution

A new explicit bijection for unicellular maps that simplifies their structure and offers bijective proofs for multiple classical formulas, including some for the first time.

Findings

Unified bijective proof for counting formulas

First bijective proof of Harer-Zagier recurrence

Extension to unicellular 3-constellations and quasi-constellations

Abstract

We consider unicellular maps, or polygon gluings, of fixed genus. A few years ago the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for these objects. In this paper, we give another bijection that explicitly describes the "recursive part" of the first bijection. As a result we obtain a very simple description of unicellular maps as pairs made by a plane tree and a permutation-like structure. All the previously known formulas follow as an immediate corollary or easy exercise, thus giving a bijective proof for each of them, in a unified way. For some of these formulas, this is the first bijective proof, e.g. the Harer-Zagier recurrence formula, the Lehman-Walsh formula and the Goupil-Schaeffer formula. We also discuss several applications of our construction: we obtain a new proof of an identity related to covered maps due to Bernardi and the first author, and thanks to previous work of the second author, we give a new expression for Stanley character polynomials, which evaluate irreducible characters of the symmetric group. Finally, we show that our techniques apply partially to unicellular 3-constellations and to related objects that we call quasi-constellations.