A bijection for covered maps on orientable surfaces

Duration: 57 mins 46 secs

About this item

Description:	Bernardi, O (CNRS) Monday 21 April 2008, 14:00-15:00 Statistical-Mechanics and Quantum-Field Theory Methods in Combinatorial Enumeration

Created:

2008-04-30 11:46

Collection:

Combinatorics and Statistical Mechanics

Publisher:

Isaac Newton Institute

Bernardi, O

Language:

eng (English)

Credits:

Producer:	Steve Greenham
Author:	Bernardi, O


Abstract:	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.

Abstract:

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.

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