2025/04/07 by Ahmia, M., Baril, J. -L., Rezig, B.
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.04828
A Catalan word of length n that avoids the pattern (≥, ≥) is a sequence w=w1⋯ wn with w1=0 and 0≤ wi≤ wi-1+1 for all i, while ensuring that no subsequence satisfies wi ≥ wi+1≥ wi+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.