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

Bijective counting of plane bipolar orientations and Schnyder woods