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

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.