Enumeration on polyominoes determined by Catalan words avoiding (≥,≥)

Abstract

A Catalan word of length n that avoids the pattern (≥,≥) is a sequence w = w ₁…w_n with w ₁ = 0 and 0 ≤ w_i ≤ w_{i −1} + 1 for all i, while ensuring that no subsequence satisfies w_i ≥ w_{i +1} ≥ w_{i +2} for i = 2, … , n. These words are enumerated by the n-th Motzkin number. From such a word, we associate a n-column Motzkin polyomino (called a (≥,≥)-polyomino), where the i-th column contains wi + 1 bottom-aligned cells. In this paper, we derive generating functions for (≥,≥)-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 (≥,≥)-polyominoes of a given length. Finally, we express all these results as linear combinations of trinomial coefficients.