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 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 $(1+x+x^2)^n$.

翻译：一个由正整数构成的词 $w=w_1\cdots w_n$ 被称为莫茨金词，当且仅当 $w_1=\texttt{1}$，对 $k=2, \dots, n$ 有 $1\leq w_k\leq w_{k-1}+1$，且 $w_{k-1}\neq w_{k}$。它可以关联到一个 $n$ 列的莫茨金多联骨牌，其中第 $i$ 列包含 $w_i$ 个单元格，且所有列均为底部对齐。我们揭示了莫茨金路径、受限卡特兰词、原始\L{}ukasiewicz 路径与莫茨金多联骨牌之间的双射关系。利用上述双射以及经典的与避免 $UDU$ 模式的 Dyck 路径的一一对应，我们给出了关于莫茨金多联骨牌的长度、面积、半周长、末符号值以及内点数的生成函数。我们针对给定长度的所有莫茨金多联骨牌，给出了总面、总半周长、末符号值之和以及总内点数的渐近结果与闭式表达式。我们还提出并证明了一个关于位于不同层级的单元格数与展开式 $(1+x+x^2)^n$ 首项之间的有趣三项式关系。