Counting planar Eulerian orientations

• Published in: European journal of combinatorics 2018-06, Vol.71, p.73-98
• Main Authors: Elvey-Price, Andrew, Guttmann, Anthony J.
• Format: Article
• Language: English
• ISSN: 0195-6698; 1095-9971
• DOI: 10.1016/j.ejc.2018.02.040

Inspired by the paper of Bonichon et al. (2016), 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=43π for 4-valent Eulerian orientations counted by vertices.