A word $w=w_1\cdots w_n$ over the set of positive integers is a Motzkin word whenever $w_1=\texttt{1}$, $1\leq w_k\leq w_{k-1}+1$, and $w_{k-1}\neq w_{k}$ 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 {\L}ukasiewicz 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 close 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 $(1+x+x^2)^n$.

翻译：一个定义于正整数集合上的单词$w=w_1\cdots w_n$被称为莫茨金单词，当且仅当$w_1=\texttt{1}$，且对于$k=2, \dots, n$，有$1\leq w_k\leq w_{k-1}+1$且$w_{k-1}\neq w_{k}$。该单词可关联至一个$n$列莫茨金多米诺，其中第$i$列包含$w_i$个单元格，且所有列均底部对齐。本文揭示了莫茨金路径、受限Catalan单词、本原Łukasiewicz路径与莫茨金多米诺之间的双射关系。利用上述双射以及经典的双射关系（与避免$UDU$子模式的Dyck路径对应），我们给出了关于莫茨金多米诺的长度、面积、半周长、末符号值以及内部点数量的生成函数。针对给定长度的所有莫茨金多米诺，我们得到总面积、总半周长、末符号值之和以及内部点总数的渐近表达式与闭合公式。此外，我们还提出并证明了一个有趣的三项式关系，该关系涉及不同层级的单元格数量与展开式$(1+x+x^2)^n$的首项。