Patterns in rectangulations. Part I: $\top$-like patterns, inversion sequence classes $I(010, 101, 120, 201)$ and $I(011, 201)$, and rushed Dyck paths
Year: 2025
Type: article
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. 30 pages, 21 figures
DOI: 10.46298/dmtcs.15118
Authors: Andrei Asinowski, Michaela A. Polley
Institutions: University of Klagenfurt, Dartmouth College
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Funder: Austrian Science Fund
OpenAlex ID: W4415267405
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