The Hopf algebra of generic rectangulations

TL;DR

This paper explores the Hopf algebra structure of generic rectangulations, establishing a connection with 2-clumped permutations and describing algebraic operations via a lattice of rectangulations.

Contribution

It introduces a lattice structure for generic rectangulations and characterizes the Hopf algebra operations through this lattice, linking combinatorics and algebra.

Findings

Describes the cover relations in the lattice of generic rectangulations.

Defines the product and coproduct operations in the Hopf algebra using the lattice.

Establishes an isomorphism between the Hopf algebra of 2-clumped permutations and generic rectangulations.

Abstract

A family of permutations called 2-clumped permutations forms a basis for a sub-Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. The 2-clumped permutations are in bijection with certain decompositions of a square into rectangles, called generic rectangulations. Thus, we can describe the Hopf algebra of 2-clumped permutations using generic rectangulations (we call this isomorphic Hopf algebra the Hopf algebra of generic rectangulations). In this paper, we describe the cover relations in a lattice of generic rectangulations that is a lattice quotient of the right weak order on permutations. We then use this lattice to describe the product and coproduct operations in the Hopf algebra of generic rectangulations.