Counting Planar Eulerian Orientations

TL;DR

This paper develops functional equations characterizing generating functions for planar Eulerian orientations and related structures, providing algorithms to compute series coefficients and conjecturing their asymptotic behavior involving constants with pi.

Contribution

It introduces a system of functional equations for counting planar Eulerian orientations and related configurations, offering polynomial-time algorithms for coefficient computation and conjecturing their asymptotic forms.

Findings

Generated 100 terms for $U(x)$ and 90 for $A(x)$.

Conjectured asymptotic behavior involving constants with pi.

Linked to the 6-vertex model in mathematical physics.

Abstract

Inspired by the paper of Bonichon, Bousquet-M\'elou, Dorbec and Pennarun, we give a system of functional equations which characterise the ordinary generating function, $U(x),$ for the number of planar Eulerian orientations counted by edges. We also characterise the ogf $A(x)$, for 4-valent planar Eulerian orientations counted by vertices in a similar way. The latter problem is equivalent to the 6-vertex problem on a random lattice, widely studied in mathematical physics. While unable to solve these functional equations, they immediately provide polynomial-time algorithms for computing the coefficients of the generating function. From these algorithms we have obtained 100 terms for $U(x)$ and 90 terms for $A(x).$ Analysis of these series suggests that they both behave as $const\cdot (1 - \mu x)/\log(1 - \mu x),$ where we conjecture that $\mu = 4\pi$ for Eulerian orientations counted by edges and $\mu=4\sqrt{3}\pi$ for 4-valent Eulerian orientations counted by vertices.