From geometry to generating functions: rectangulations and permutations

TL;DR

This paper explores the enumeration of pattern-avoiding rectangulations, establishes bijections with permutations, proves algebraic generating functions, and introduces a new class called whirls, advancing combinatorial understanding.

Contribution

It introduces new bijections between rectangulations and permutations, proves algebraicity of their generating functions, and analyzes a novel rectangulation class called whirls.

Findings

Generated algebraic functions for pattern-avoiding rectangulations.

Established bijections linking rectangulations and permutations.

Analyzed the structure of whirls using generating trees.

Abstract

We enumerate several classes of pattern-avoiding rectangulations. We establish new bijective links with pattern-avoiding permutations, prove that their generating functions are algebraic, and confirm several conjectures by Merino and M\"utze. We also analyze a new class of rectangulations, called whirls, using a generating tree.