Plane bipolar orientations and quadrant walks

Understand

Bipolar orientations of planar maps have recently attracted some interest in combinatorics, probability theory and theoretical physics.

• Plane bipolar orientations with $n$ edges are known to be counted by the $n$th Baxter number $b(n)$, which can be defined by a linear recurrence relation with polynomial coefficients.
• Equivalently, the associated generating function $\sum_n b(n)t^n$ is D-finite.
• In this paper, we address a much refined enumeration problem, where we record for every $r$ the number of faces of degree $r$.