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The generating function of planar Eulerian orientations

Mireille Bousquet-Mélou, Andrew Elvey Price

Abstract

The enumeration of planar maps equipped with an Eulerian orientation has attracted attention in both combinatorics and theoretical physics since at least 2000. The case of 4-valent maps is particularly interesting: these orientations are in bijection with properly 3-coloured quadrangulations, while in physics they correspond to configurations of the ice model. We solve both problems -- namely the enumeration of planar Eulerian orientations and of 4-valent planar Eulerian orientations -- by expressing the associated generating functions as the inverses (for the composition of series) of simple hypergeometric series. Using these expressions, we derive the asymptotic behaviour of the number of planar Eulerian orientations, thus proving earlier predictions of Kostov, Zinn-Justin, Elvey Price and Guttmann. This behaviour, $μ^n /(n \log n)^2$, prevents the associated generating functions from being D-finite. Still, these generating functions are differentially algebraic, as they satisfy non-linear differential equations of order $2$. Differential algebraicity has recently been proved for other map problems, in particular for maps equipped with a Potts model. Our solutions mix recursive and bijective ingredients. In particular, a preliminary bijection transforms our oriented maps into maps carrying a height function on their vertices. In the 4-valent case, we also observe an unexpected connection with the enumeration of maps equipped with a spanning tree that is internally inactive in the sense of Tutte. This connection remains to be explained combinatorially.

The generating function of planar Eulerian orientations

Abstract

The enumeration of planar maps equipped with an Eulerian orientation has attracted attention in both combinatorics and theoretical physics since at least 2000. The case of 4-valent maps is particularly interesting: these orientations are in bijection with properly 3-coloured quadrangulations, while in physics they correspond to configurations of the ice model. We solve both problems -- namely the enumeration of planar Eulerian orientations and of 4-valent planar Eulerian orientations -- by expressing the associated generating functions as the inverses (for the composition of series) of simple hypergeometric series. Using these expressions, we derive the asymptotic behaviour of the number of planar Eulerian orientations, thus proving earlier predictions of Kostov, Zinn-Justin, Elvey Price and Guttmann. This behaviour, , prevents the associated generating functions from being D-finite. Still, these generating functions are differentially algebraic, as they satisfy non-linear differential equations of order . Differential algebraicity has recently been proved for other map problems, in particular for maps equipped with a Potts model. Our solutions mix recursive and bijective ingredients. In particular, a preliminary bijection transforms our oriented maps into maps carrying a height function on their vertices. In the 4-valent case, we also observe an unexpected connection with the enumeration of maps equipped with a spanning tree that is internally inactive in the sense of Tutte. This connection remains to be explained combinatorially.

Paper Structure

This paper contains 17 sections, 26 theorems, 111 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Planar maps
  4. Orientations
  5. The 6-vertex model and fully packed loops
  6. Formal power series
  7. Functional equations for quartic Eulerian orientations
  8. Solution for quartic Eulerian orientations
  9. A bijection
  10. Functional equations for general Eulerian orientations
  11. Solution for general Eulerian orientations
  12. Nature of the series and asymptotics
  13. Nature of the series
  14. Asymptotics
  15. Final comments and perspectives
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let ${\sf R}(t)\equiv {\sf R}$ be the unique formal power series with constant term $0$ satisfying Then the generating function of quartic rooted planar Eulerian orientations, counted by vertices, is This is a differentially algebraic series, satisfying a non-linear differential equation of order $2$ whose coefficients are polynomials in $t$. The number $q_n$ of such orientations having $n$ vert

Figures (25)

  • Figure 1: Left: a rooted planar map, which is 4-valent (or: quartic). Right: the same map, equipped with an Eulerian orientation.
  • Figure 2: The planar Eulerian orientations with at most two edges, in agreement with ${\sf G}(t)=t+5t^2+O(t^3)$. On the right are the four quartic Eulerian orientations with one vertex, in agreement with ${\sf Q}(t)=4t+O(t^2)$.
  • Figure 3: Left: a rooted planar map, with the root edge and root corner shown. Right: the dual map, in dashed edges.
  • Figure 4: A labelled map.
  • Figure 5: An Eulerian orientation (solid edges) and the corresponding dual labelled map (dashed edges). The labelling rule is shown on the right.
  • ...and 20 more figures

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 39 more