Algebraic and combinatorial structures on Baxter permutations

TL;DR

This paper introduces a new Hopf algebra structure on Baxter permutations and related objects, using a novel Baxter monoid and an insertion algorithm, expanding algebraic understanding of Baxter combinatorics.

Contribution

It constructs a new Hopf subalgebra of Free quasi-symmetric functions based on Baxter permutations, with a novel Baxter monoid and insertion algorithm.

Findings

Defined the Baxter monoid as an analog of plactic and sylvester monoids.

Constructed a Hopf subalgebra with bases indexed by Baxter objects.

Studied algebraic properties of the new Hopf algebra.

Abstract

We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e. Baxter permutations, pairs of twin binary trees, etc.). This construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-Schensted-like insertion algorithm. The algebraic properties of this Hopf algebra are studied. This Hopf algebra appeared for the first time in the work of Reading [Lattice congruences, fans and Hopf algebras, Journal of Combinatorial Theory Series A, 110:237--273, 2005].