Bijective Counting of Involutive Baxter Permutations

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 ${3\cdot2^{n-1}\over (n+1)(n+2)} \big({2n \atop n}\big)$, a formula originally discovered by M. Bousquet-Mélou 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.