Generic rectangulations

TL;DR

This paper introduces a new combinatorial approach to enumerate generic rectangulations by establishing a bijection with a specific class of pattern-avoiding permutations.

Contribution

It provides the first explicit bijection between generic rectangulations and a class of pattern-avoiding permutations, advancing the enumeration of these tilings.

Findings

Established a bijection between generic rectangulations and pattern-avoiding permutations.

Initiated the enumeration of generic rectangulations up to combinatorial equivalence.

Connected rectangulation enumeration to permutation pattern avoidance.

Abstract

A rectangulation is a tiling of a rectangle by a finite number of rectangles. The rectangulation is called generic if no four of its rectangles share a single corner. We initiate the enumeration of generic rectangulations up to combinatorial equivalence by establishing an explicit bijection between generic rectangulations and a set of permutations defined by a pattern-avoidance condition analogous to the definition of the twisted Baxter permutations.