The Hopf algebra of diagonal rectangulations

TL;DR

This paper introduces a combinatorial Hopf algebra based on diagonal rectangulations, linking it to twisted Baxter permutations and related polytope structures.

Contribution

It provides an intrinsic combinatorial realization of the Hopf algebra tBax and explores its connections to lattice structures and polytopes.

Findings

Defined the Hopf algebra dRec with basis elements as diagonal rectangulations.

Established a bijection between twisted Baxter and Baxter permutations.

Connected diagonal rectangulations to a polytope analogous to the associahedron.

Abstract

We define and study a combinatorial Hopf algebra dRec with basis elements indexed by diagonal rectangulations of a square. This Hopf algebra provides an intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter permutations, which previously had only been described extrinsically as a sub Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. We describe the natural lattice structure on diagonal rectangulations, analogous to the Tamari lattice on triangulations, and observe that diagonal rectangulations index the vertices of a polytope analogous to the associahedron. We give an explicit bijection between twisted Baxter permutations and the better-known Baxter permutations, and describe the resulting Hopf algebra structure on Baxter permutations.