Bijective counting of plane bipolar orientations

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. Writing ϑij for the number of plane bipolar orientations with (i+1) vertices and (j+1) faces, our bijection provides a combinatorial proof of the following formula due to Baxter:(1)ϑij=2(i+j−2)!(i+j−1)!(i+j)!(i−1)!i!(i+1)!(j−1)!j!(j+1)!.