Baxter permutations and plane bipolar orientations

TL;DR

This paper introduces a simple bijection linking Baxter permutations to plane bipolar orientations, translating permutation parameters into orientation features, and extends to special classes relating to non-separable and series-parallel maps.

Contribution

It provides a novel bijection connecting Baxter permutations with plane bipolar orientations and their subclasses, revealing symmetry properties and combinatorial correspondences.

Findings

Baxter permutations correspond to plane bipolar orientations with n edges.

Specializations relate Baxter permutations avoiding certain patterns to specific map classes.

The bijection preserves and translates key combinatorial parameters.

Abstract

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 (number of ascents, right-to-left maxima, left-to-right minima...) into natural parameters of plane bipolar orientations (number of vertices, degree of the sink, degree of the source...), and has remarkable symmetry properties. By specializing it to Baxter permutations avoiding the pattern 2413, we obtain a bijection with non-separable planar maps. A further specialization yields a bijection between permutations avoiding 2413 and 3142 and series-parallel maps.