Baxter permutations and plane bipolar orientations

We present a simple bijection between Baxter permutations of size n and plane bipolar orientations with n edges. This bijection translates several classical parameters of permutations into natural parameters of orientations, and has remarkable symmetry properties. By specialising it to Baxter permutations avoiding the pattern 3142, we obtain a bijection with non-separable planar maps, which had been described only in an implicit recursive manner so far (up to simple symmetries). A further specialization yields a bijection between permutations avoiding 3142 and 2413 and series-parallel maps.