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：令正整数集上的单词$w=w_1\cdots w_n$满足：$w_1=\texttt{1}$，对于$k=2, \dots, n$有$1\leq w_k\leq w_{k-1}+1$且$w_{k-1}\neq w_{k}$，则称$w$为莫茨金单词。该单词可关联至一个$n$列莫茨金多联骨牌，其第$i$列包含$w_i$个单元格，且所有列底端对齐。我们揭示了莫茨金路径、受限Catalan单词、原始Łukasiewicz路径与莫茨金多联骨牌之间的双射联系。借助上述双射以及经典的避免$UDU$的Dyck路径一一对应关系，我们给出了关于莫茨金多联骨牌的长度、面积、半周长、末符号取值及内部点数量的生成函数。针对给定长度的所有莫茨金多联骨牌，我们给出了总面积、总半周长、末符号值之和及内部点总数的渐近式与封闭表达式。我们还提出并证明了一个涉及不同层级单元格数量与展开式$(1+x+x^2)^n$首项的有趣三项式关系。