Enumeration on polyominoes determined by Catalan words avoiding $(\geq,\geq)$

TL;DR

This work studies polyominoes derived from Catalan words of length $n$ that avoid the pattern $(\geq, \geq)$, a set counted by the Motzkin numbers $m_n$. By associating each word with a $(\geq, \geq)$-polyomino and employing multivariate generating functions, the authors derive exact and asymptotic results for several statistics: area, semiperimeter, the last-symbol value, and the number of interior points. Key contributions include closed-form generating functions, kernel-method solutions, and intricate connections to central trinomial coefficients $T_n$ and Motzkin numbers, with several results expressed as linear combinations of trinomial terms. A central theme is the translation of statistics on $(\geq, \geq)$-Catalan polyominoes into tractable combinatorial identities via bijections to $(\neq)$-Catalan polyominoes and established trinomial frameworks. These results extend prior work on Catalan polyominoes and pattern-avoiding words, providing a cohesive toolkit for enumerating and analyzing these constrained polyomino families and highlighting deep links to Motzkin and trinomial combinatorics.

Abstract

A Catalan word of length $n$ that avoids the pattern $(\geq, \geq)$ is a sequence $w=w_1\cdots w_n$ with $w_1=0$ and $0\leq w_i\leq w_{i-1}+1$ for all $i$, while ensuring that no subsequence satisfies $w_i \geq w_{i+1}\geq w_{i+2}$ for $i=2,\ldots,n$. These words are enumerated by the $n$-th Motzkin number. From such a word, we associate a $n$-column Motzkin polyomino (called a $(\geq,\geq)$-polyomino), where the $i$-th column contains $w_i+1$ bottom-aligned cells. In this paper, we derive generating functions for $(\geq,\geq)$-polyominoes based on their length, area, semiperimeter, last symbol value, and number of interior points. We provide asymptotic analyses and closed-form expressions for the total area, total semiperimeter, sum of the last symbol values, and total number of interior points across all $(\geq,\geq)$-polyominoes of a given length. Finally, we express all these results as linear combinations of trinomial coefficients.

Figures (9)

• Figure 1: Dyck path associated with the Catalan word $\texttt{00123223401011}$, and its corresponding polyomino.
• Figure 2: All $(\geq, \geq)$-polyominoes of length $4$.
• Figure 3: Case (i): The last column is of height $1$.
• Figure 4: Case (ii): The last column is of height at least $2$.
• Figure 5: The $6$$(\geq,\geq)$-polyominoes of length $4$ and semiperimeter $7$.

Theorems & Definitions (26)

• Theorem 2.1
• proof
• Corollary 2.2
• Corollary 2.3
• Corollary 2.4
• proof
• Theorem 3.1
• Theorem 3.2
• Corollary 3.3
• Corollary 3.4