Bijective counting of plane bipolar orientations

Eric Fusy (1) , Dominique Poulalhon (2) , Gilles Schaeffer (1)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

We introduce a bijection between plane bipolar orientations with fixed numbers of vertices and faces, and non-intersecting triples of upright lattice paths with some specific extremities. It provides a combinatorial proof of a nice formula due to Baxter for the number of plane bipolar orientations with $(i+1)$ vertices and $(j+1)$ faces.

Domaines: Combinatoire [math.CO]

Eric Fusy, Dominique Poulalhon, Gilles Schaeffer. Bijective counting of plane bipolar orientations. European Conference on Combinatorics, Graph Theory and Applications (EuroComb), 2007, Malaga, Spain. pp.283-287. ⟨hal-00160430⟩