The combinatorics of Motzkin polyominoes

Abstract

A word $w=w_1\cdots w_n$ over the set of positive integers is a Motzkin word whenever $w_1=1$, $1\le w_k\le w_{k-1}+1$, and $w_{k-1}\ne w_k$ for $k=2,\dots ,n$. It can be associated to a $n$-column Motzkin polyomino whose $i$-th column contains $w_i$ cells, and all columns are bottom-justified. We reveal bijective connections between Motzkin paths, restricted Catalan words, primitive Łukasiewicz paths, and Motzkin polyominoes. Using the aforementioned bijections together with classical one-to-one correspondence with Dyck paths avoiding $UDU$s, we provide generating functions with respect to the length, area, semiperimeter, value of the last symbol, and number of interior points of Motzkin polyominoes. We give asymptotics and closed-form expressions for the total area, total semiperimeter, sum of the last symbol values, and total number of interior points over all Motzkin polyominoes of a given length. We also present and prove an engaging trinomial relation concerning the number of cells lying at different levels and first terms of the expanded ${(1+x+x^2)}^n$.