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arXiv · 0803.0400

Bijective counting of plane bipolar orientations and Schnyder woods

Eric Fusy·Dominique Poulalhon·Gilles Schaeffer
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Abstract

A bijection $Φ$ is presented between plane bipolar orientations with prescribed numbers of vertices and faces, and non-intersecting triples of upright lattice paths with prescribed extremities. This yields a combinatorial proof of the following formula due to R. Baxter for the number $Θ_{ij}$ of plane bipolar orientations with $i$ non-polar vertices and $j$ inner faces: $Θ_{ij}=2\frac{(i+j)!(i+j+1)!(i+j+2)!}{i!(i+1)!(i+2)!j!(j+1)!(j+2)!}$. In addition, it is shown that $Φ$ specializes into the bijection of Bernardi and Bonichon between Schnyder woods and non-crossing pairs of Dyck words.

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Eric Fusy, Dominique Poulalhon, Gilles Schaeffer. 2009-03-20. Bijective counting of plane bipolar orientations and Schnyder woods. https://arxiv.org/abs/0803.0400

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