Counting planar Eulerian orientations

Abstract

Inspired by the paper of Bonichon, Bousquet-Mélou, Dorbec and Pennarun [1], 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·(1−µx)/ log(1− µx), where we conjecture that µ = 4π for Eulerian orientations counted by edges and µ = 4 √ 3π for 4-valent Eulerian orientations counted by vertices.arXiv:1707.09120v3 [math.CO]