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 ϑ i j 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) ϑ i j = 2 ( i + j − 2 ) ! ( i + j − 1 ) ! ( i + j ) ! ( i − 1 ) ! i ! ( i + 1 ) ! ( j − 1 ) ! j ! ( j + 1 ) ! .