Bijective counting of involutive Baxter permutations

TL;DR

This paper provides a bijective enumeration of involutive Baxter permutations, offering an elementary proof for their count with no fixed points, and connects these permutations to planar maps with specific orientations.

Contribution

It introduces a bijective approach to count involutive Baxter permutations and links their enumeration to planar maps with acyclic orientations and unique sources.

Findings

Number of involutive Baxter permutations of size 2n with no fixed points is given by a specific formula.

The same enumeration formula applies to certain planar maps with acyclic orientations.

Elementary bijective proof replaces previous generating function methods.

Abstract

We enumerate bijectively the family of involutive Baxter permutations according to various parameters; in particular we obtain an elementary proof that the number of involutive Baxter permutations of size $2n$ with no fixed points is $\frac{3\cdot 2^{n-1}}{(n+1)(n+2)}\binom{2n}{n}$, a formula originally discovered by M. Bousquet-M\'elou using generating functions. The same coefficient also enumerates planar maps with $n$ edges, endowed with an acyclic orientation having a unique source, and such that the source and sinks are all incident to the outer face.