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Combinatorial proof for the rationality of the bivariate generating series of maps in positive genus

Marie Albenque, Mathias Lepoutre

Abstract

In this paper, we give the first combinatorial proof of a rationality scheme for the generating series of maps in positive genus enumerated by both vertices and faces, which was first obtained by Bender, Canfield and Richmond in 1993 by purely computational techniques. To do so, we rely on a bijection obtained by the second author in a previous work between those maps and a family of decorated unicellular maps. Our main contribution consists in a fine analysis of this family of maps. As a byproduct, we also obtain a new and simpler combinatorial proof of the rationality scheme for the generating series of maps enumerated by their number of edges, originally obtained computationally by Bender and Canfield in 1991 and combinatorially by the second author in 2019.

Combinatorial proof for the rationality of the bivariate generating series of maps in positive genus

Abstract

In this paper, we give the first combinatorial proof of a rationality scheme for the generating series of maps in positive genus enumerated by both vertices and faces, which was first obtained by Bender, Canfield and Richmond in 1993 by purely computational techniques. To do so, we rely on a bijection obtained by the second author in a previous work between those maps and a family of decorated unicellular maps. Our main contribution consists in a fine analysis of this family of maps. As a byproduct, we also obtain a new and simpler combinatorial proof of the rationality scheme for the generating series of maps enumerated by their number of edges, originally obtained computationally by Bender and Canfield in 1991 and combinatorially by the second author in 2019.

Paper Structure

This paper contains 37 sections, 24 theorems, 104 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Notations, definitions and first constructions
  3. Notations
  4. Maps
  5. Bicolorable maps and the radial construction
  6. Unicellular maps, cores, schemes
  7. Orientations
  8. Bijection for higher-genus maps, counting vertices and faces
  9. Definition of blossoming maps
  10. Closure of blossoming maps
  11. First definition of closure
  12. Closure with labels
  13. Closure and orientations
  14. Good blossoming maps and closures
  15. Analysis of good maps
  16. ...and 22 more sections

Key Result

Theorem \oldthetheorem

Let $T(z)$ be the unique formal power series in $z$ that satisfies $T(z)=z+3T^2(z)$ and $T(0)=0$. Then, for any $g\geq 0$, $M_g(z)$ is a rational function of $T(z)$.

Figures (18)

  • Figure 1: Illustration of the bijection of Lep19 for a 4-valent map on a torus of genus 1. We associate to a map (represented in \ref{['subfig:Genus1Intro']}) one of its spanning unicellular submaps decorated by some opening and closing half-edges (respresented in \ref{['subfig:BlossomingIntro']}). These half-edges can be matched as in a parenthis word to reconstruct the facial cycles.
  • Figure 2: Two embeddings of a connected graph on the torus of genus 1. On \ref{['subfig:notMaps']}, the embedding is not cellular: the shaded face is indeed not homeomorphic to a disk. On \ref{['subfig:MapsTore']}, the embedding is cellular and defines a map of genus 1.
  • Figure 3: Illustration of the radial construction. The original map is represented with blue edges and square vertices and its radial by black edges and round vertices. In \ref{['subfig:radial1']} and \ref{['subfig:radial2']}, we use the classical representation of the torus of genus 1 as a square where both pairs of its opposite sides must be identified. In \ref{['subfig:radial2']}, the faces of the radial map are colored in black (actually gray for sake of visibility) and white, following our coloring convention.
  • Figure 4: In \ref{['subfig:coreDec']}, we represent schematically the decomposition of a unicellular map into a forest of trees (represented in thin black edges) and a core (represented in fat blue edges). Its scheme is represented in \ref{['subfig:schemeHexa']}. In \ref{['subfig:schemeCarre']}, we represent the other possible scheme of genus 1.
  • Figure 5: The same 4-valent bicolorable map endowed with some Eulerian orientations with no clockwise face. The one in \ref{['subfig:dualgeodheight']} is bicolorable and corresponds to the dual-geodesic orientation, the one in \ref{['subfig:noBicolor']} is not bicolorable.
  • ...and 13 more figures

Theorems & Definitions (67)

  • Theorem \oldthetheorem: Tutte Tut63 for $g=0$, Bender and Canfield BenCan91 for $g\geq 1$
  • Theorem \oldthetheorem: Arquès Arq85Arq87 for $g=0,1$, Bender, Canfield and Richmond BeCaRi93 for $g\geq 2$
  • Theorem \oldthetheorem: BeCaRi93ArqGio99
  • Proposition \oldthetheorem
  • Remark 2.1
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Remark 2.2
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 57 more