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Counting Eulerian orientations Guttmann, Tony


Inspired by the paper of Bonichon, Bousquet-M\'elou, Dorbec and Pennarun, we give a system of functional equations for the ogfs for the number of planar Eulerian orientations counted by the edges, $U(x),$ and the number of 4-valent planar Eulerian orientations counted by the number of vertices, $A(x)$. 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 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 $ \cdot (1 - \mu x)^2/\log(1 - \mu x),$ where we make the confident conjectures that $\mu = 4\pi$ for Eulerian orientations counted by edges and $\mu=4\sqrt{3}\pi$ for 4-valent Eulerian orientations counted by vertices. (Joint work with Andrew Elvey Price).

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