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 work develops a formal framework for pattern avoidance in rectangulations, focusing on <tr>-like patterns and enumerating L-avoiding classes across all rotations. It shows that weak <tr>-avoidance yields Catalan-number enumeration and strong avoidance maps to inversion-sequence classes, including $I(010,101,120,201)$ and $I(011,201)$, while $\\\ abla$-avoiding strong rectangulations relate to rushed Dyck paths and related Dyck-path families. The authors establish Wilf-equivalences among multiple inversion-sequence classes, provide explicit bijections to Catalan structures, compositions, and rushed Dyck paths, and lay a comprehensive groundwork for representing rectangulation patterns via permutation patterns in future work. These results expose rich connections among rectangulations, inversion sequences, and Dyck-path families, offering a path toward a systematic theory of pattern avoidance in rectangulations with potential algorithmic and enumerative applications.

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.

Figures (21)

• Figure 1: Four rectangulations of size $9$. Rectangulations $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are weakly equivalent. Rectangulations $\mathcal{A}$ and $\mathcal{B}$ are also strongly equivalent. Rectangulation $\mathcal{B}$ is diagonal. For rectangulations $\mathcal{A}$ and $\mathcal{B}$, the NW--SE labeling is shown.
• Figure 2: Plot of the inversion sequence $e=(0,0,0,3,4,3,5)$.
• Figure 3: Images (a) and (b) represent the same strong pattern, and (c) a different strong pattern. All three images represent the same weak pattern.
• Figure 4: Two drawings of the same pattern (a,b), and its occurrence in a rectangulation (c).
• Figure 5: The structural decomposition of an element of $R^w(<tr>)$.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• proof
• Proposition 5
• Theorem 6
• proof
• Proposition 7