Combinatorics of rectangulations: Old and new bijections

TL;DR

This paper explores the combinatorial structures of rectangulations, establishing bijections with permutation classes, characterizing special cases like guillotine rectangulations through pattern avoidance, and providing enumeration bounds.

Contribution

It unifies and simplifies bijections between rectangulations and permutations, introduces new characterizations of guillotine rectangulations, and addresses enumeration of strong rectangulation families.

Findings

Unified treatment of weak and strong rectangulation bijections

Characterization of guillotine rectangulations via pattern avoidance

Asymptotic bounds for enumeration of strong rectangulations

Abstract

A rectangulation is a decomposition of a rectangle into finitely many rectangles. Via natural equivalence relations, rectangulations can be seen as combinatorial objects with a rich structure, with links to lattice congruences, flip graphs, polytopes, lattice paths, Hopf algebras, etc. In this paper, we first revisit the structure of the respective equivalence classes: weak rectangulations that preserve rectangle-segment adjacencies, and strong rectangulations that preserve rectangle-rectangle adjacencies. We thoroughly investigate posets defined by adjacency in rectangulations of both kinds, and unify and simplify known bijections between rectangulations and permutation classes. This yields a uniform treatment of mappings between permutations and rectangulations that unifies the results from earlier contributions, and emphasizes parallelism and differences between the weak and the strong cases. Then, we consider the special case of guillotine rectangulations, and prove that they can be characterized - under all known mappings between permutations and rectangulations - by avoidance of two mesh patterns that correspond to "windmills" in rectangulations. This yields new permutation classes in bijection with weak guillotine rectangulations, and the first known permutation class in bijection with strong guillotine rectangulations. Finally, we address enumerative issues and prove asymptotic bounds for several families of strong rectangulations.