A bijection for covered maps, or a shortcut between Harer-Zagier's and Jackson's formulas
Résumé
We consider maps on orientable surfaces. A map is unicellular if it has a single face. A covered map is a map with a marked unicellular spanning submap. For a map of genus g, the unicellular submap can have any genus g'=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, the covered maps are maps with a marked spanning tree (a.k.a. tree-rooted maps) and our bijection specializes into a construction previously described by the first author. A strong connection subsists between covered maps and tree-rooted maps in genus 1 (because a covered map is either a tree-rooted map or the dual of a tree-rooted map) and we thereby obtain a bijective explanation of a formula by Lehman and Walsh on the number of tree-rooted maps of genus 1. A more surprising byproduct of our bijection is an equivalence between an enumerative formula by Harer and Zagier concerning unicellular maps of given genus and a similar formula by Jackson concerning bipartite unicellular maps of given genus. The equivalence is obtained by observing that covered maps can be seen as a shuffle of two unicellular maps, hence that our bijection gives a relations between shuffles of unicellular maps and bipartite unicellular maps. > > We also show that the bijection of Bouttier, Di Francesco and Guitter (which generalizes a famous bijection by Schaeffer) between bipartite maps and so-called well-labelled mobiles can be described as a special case of our bijection.
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