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

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.

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