Patterns in rectangulations. Part I: $\top$-like patterns, inversion sequence classes $I(010, 101, 120, 201)$ and $I(011, 201)$, and rushed Dyck paths

TL;DR

This paper systematically studies pattern avoidance in rectangulations, establishing enumerations, bijections with Catalan structures, inversion sequences, and rushed Dyck paths, revealing new combinatorial correspondences.

Contribution

It introduces formal definitions of pattern avoidance in rectangulations and establishes novel bijections with Catalan structures, inversion sequences, and rushed Dyck paths.

Findings

Catalan numbers enumerate $ op$-avoiding weak rectangulations.

Bijections are constructed between $ op$-avoiding strong rectangulations and inversion sequences.

Strong rectangulations avoiding $ op$ and $ot$ are bijective with rushed Dyck paths.

Abstract

We initiate a systematic study of pattern avoidance in rectangulations. We give a formal definition of such patterns and investigate rectangulations that avoid $\top$-like patterns - the pattern $\top$ and its rotations. For every $L \subseteq \{\top, \, \vdash, \, \bot, \, \dashv \}$ we enumerate $L$-avoiding rectangulations, both weak and strong. In particular, we show $\top$-avoiding weak rectangulations are enumerated by Catalan numbers and construct bijections to several Catalan structures. Then, we prove that $\top$-avoiding strong rectangulations are in bijection with several classes of inversion sequences, among them $I(010,101,120,201)$ and $I(011,201)$ - which leads to a solution of the conjecture that these classes are Wilf-equivalent. Finally, we show that $\{\top, \bot\}$-avoiding strong rectangulations are in bijection with recently introduced rushed Dyck paths.