Bijective counting of plane bipolar orientations and Schnyder woods

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 Θ i j of plane bipolar orientations with i non-polar vertices and j inner faces: Θ i j = 2 ( 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. s 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).