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Bijective counting of plane bipolar orientations and Schnyder woods
Authors:   ric Fusy, Dominique Poulalhon,Gilles Schaeffer
Affiliation:aLIX, École Polytechnique, 91128 Palaiseau Cedex, France;bLIAFA, Université Paris Diderot, case 7014, 75205 Paris Cedex 13, France
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 Baxter for the number Θij of plane bipolar orientations with i non-polar vertices and j inner faces:
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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.This is the extended and revised journal version of a conference paper with the title “Bijective counting of plane bipolar orientations”, which appeared in Electr. Notes in Discr. Math. pp. 283–287 (Proceedings of Eurocomb’07, 11–15 September 2007, Sevilla).
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