A bijection for covered maps on orientable surfaces
Seminar Room 1, Newton Institute
A map of genus g is a graph together with an embedding in the orientable surface of genus g. For instance, plane trees can be considered as maps of genus 0 and unicellular maps (maps with a single face) are a natural generalisation of plane trees to higher genus surfaces.
In this talk, we consider covered maps, which are maps together with a distinguished unicellular spanning submap. We will present a bijection between covered maps of genus g with n edges and pairs made of a plane tree with n edges and a bipartite unicellular map of genus g with n+1 edges. This bijection allows to recover bijectively some very elegant formulas by Mullin and by Lehman and Walsh. We will also show that our bijection generalises a bijection of Bouttier, Di Francesco and Guitter (which, in turns, generalises a previous bijection of Schaeffer) between bipartite maps and some classes of labelled trees.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.