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On the number of planar Eulerian orientations

Nicolas Bonichon, Mireille Bousquet-Mélou, Paul Dorbec, Claire Pennarun

TL;DR

The paper tackles the challenging problem of counting planar Eulerian orientations by introducing k-indexed subsets and supersets that converge to the full set as k grows. It develops two recursive decompositions (standard and prime) and shows that the associated generating functions are algebraic, using polynomial systems and Popescu’s approximation theorem for the divided-difference cases. Substantial results include explicit algebraic systems and growth-rate bounds that suggest a growth constant near 12.5, with progressively tighter bounds from both the subset and superset constructions. The work provides a robust framework for bounding and understanding Eulerian orientations on planar maps, with explicit small-k cases, asymptotic analyses, and several open questions about convergence of bounds and universality of the singular behavior.

Abstract

The number of planar Eulerian maps with n edges is well-known to have a simple expression. But what is the number of planar Eulerian orientations with n edges? This problem appears to be difficult. To approach it, we define and count families of subsets and supersets of planar Eulerian orientations, indexed by an integer k, that converge to the set of all planar Eulerian orientations as k increases. The generating functions of our subsets can be characterized by systems of polynomial equations, and are thus algebraic. The generating functions of our supersets are characterized by polynomial systems involving divided differences, as often occurs in map enumeration. We prove that these series are algebraic as well. We obtain in this way lower and upper bounds on the growth rate of planar Eulerian orientations, which appears to be around 12.5.

On the number of planar Eulerian orientations

TL;DR

The paper tackles the challenging problem of counting planar Eulerian orientations by introducing k-indexed subsets and supersets that converge to the full set as k grows. It develops two recursive decompositions (standard and prime) and shows that the associated generating functions are algebraic, using polynomial systems and Popescu’s approximation theorem for the divided-difference cases. Substantial results include explicit algebraic systems and growth-rate bounds that suggest a growth constant near 12.5, with progressively tighter bounds from both the subset and superset constructions. The work provides a robust framework for bounding and understanding Eulerian orientations on planar maps, with explicit small-k cases, asymptotic analyses, and several open questions about convergence of bounds and universality of the singular behavior.

Abstract

The number of planar Eulerian maps with n edges is well-known to have a simple expression. But what is the number of planar Eulerian orientations with n edges? This problem appears to be difficult. To approach it, we define and count families of subsets and supersets of planar Eulerian orientations, indexed by an integer k, that converge to the set of all planar Eulerian orientations as k increases. The generating functions of our subsets can be characterized by systems of polynomial equations, and are thus algebraic. The generating functions of our supersets are characterized by polynomial systems involving divided differences, as often occurs in map enumeration. We prove that these series are algebraic as well. We obtain in this way lower and upper bounds on the growth rate of planar Eulerian orientations, which appears to be around 12.5.

Paper Structure

This paper contains 37 sections, 12 theorems, 91 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Recursive decompositions of Eulerian orientations
  3. Definitions
  4. Eulerian maps: standard decomposition
  5. Eulerian orientations: standard decomposition
  6. Prime decomposition of maps and orientations
  7. Subsets of Eulerian orientations, via the standard decomposition
  8. An algebraic system for $\boldsymbol\mathcal{L}^{(k)}$
  9. Examples
  10. When $\boldsymbol{k=1}$,
  11. When $\boldsymbol{k=2}$,
  12. Asymptotic analysis for subsets of Eulerian orientations
  13. Back to examples
  14. Subsets of Eulerian orientations, via the prime decomposition
  15. An algebraic system for $\boldsymbol\mathbb{L}^{(k)}$
  16. ...and 22 more sections

Key Result

Proposition 3

Consider the collection of equations consisting of: In this collection, replace all trivial $K$-series by their value: $K_{\rm{\bf w}}=0$ when ${\rm{\bf w}}$ is not balanced, $K_\varepsilon=1$. Let $S_0$ denote the resulting system. The number of series it involves is The system $S_0$ defines uniquely these $f(k)$ series. Its size can be (roughly) divided by two upon noticing that replacing all

Figures (8)

  • Figure 1: A rooted Eulerian map and a rooted Eulerian orientation.
  • Figure 2: Left: First values of $o_n$, for $n$ from 0 to 15 (entry A277493 of the OEIS oeis). Right: A plot of $o_{n+1}/o_n$ vs. $1/n$, for $n=4, \ldots, 14$, suggests that the growth rate of Eulerian orientations, located at the intercept of the curve and the $y$-axis, is around $12.5$.
  • Figure 3: Construction of an Eulerian map with $n$ edges: merge an ordered pair of Eulerian maps $M_1$, $M_2$ with $n_1$ and $n_2$ edges ($n_1 + n_2 =n-1$) and add a loop, or make a split on an Eulerian map with $n-1$ edges. The new edge (here thicker) is the root edge of $M$.
  • Figure 4: Construction of an Eulerian orientation: merge two Eulerian orientations (the loop can be oriented in either way), or split (legally) an Eulerian orientation. Observe how the root word changes.
  • Figure 5: Decomposition of an Eulerian map $M$ into prime Eulerian maps $M_1$, $M_2$, $M_3$.
  • ...and 3 more figures

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Proposition 3
  • proof
  • Remark 4
  • Proposition 5
  • proof
  • Definition 6
  • Proposition 7
  • proof
  • ...and 24 more