A bijection for covered maps, or a shortcut between Harer-Zagier's and Jackson's formulas

TL;DR

This paper introduces a bijection linking covered maps with pairs of plane trees and bipartite unicellular maps, unifying Harer-Zagier's and Jackson's formulas and connecting various map enumeration techniques.

Contribution

It presents a new bijection for covered maps that unifies existing formulas and generalizes previous bijections between maps and mobiles.

Findings

Establishes a bijection between covered maps and pairs of trees and bipartite maps.

Shows the bijection specializes to known constructions in the planar case.

Connects Harer-Zagier's and Jackson's formulas through this bijection.

Abstract

We consider maps on orientable surfaces. A map is called \emph{unicellular} if it has a single face. A \emph{covered map} is a map (of genus $g$) with a marked unicellular spanning submap (which can have any genus in $\{0,1,...,g\}$). Our main result is a bijection between covered maps with $n$ edges and genus $g$ and pairs made of a plane tree with $n$ edges and a unicellular bipartite map of genus $g$ with $n+1$ edges. In the planar case, covered maps are maps with a marked spanning tree and our bijection specializes into a construction obtained by the first author in \cite{OB:boisees}. Covered maps can also be seen as \emph{shuffles} of two unicellular maps (one representing the unicellular submap, the other representing the dual unicellular submap). Thus, our bijection gives a correspondence between shuffles of unicellular maps, and pairs made of a plane tree and a unicellular bipartite map. In terms of counting, this establishes the equivalence between a formula due to Harer and Zagier for general unicellular maps, and a formula due to Jackson for bipartite unicellular maps. We also show that the bijection of Bouttier, Di Francesco and Guitter \cite{BDFG:mobiles} (which generalizes a previous bijection by Schaeffer \cite{Schaeffer:these}) between bipartite maps and so-called well-labelled mobiles can be obtained as a special case of our bijection.