The Combinatorics of Motzkin Polyominoes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916296829960192 |
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| author | Baril, Jean-Luc Kirgizov, Sergey Ramírez, José L. Villamizar, Diego |
| author_facet | Baril, Jean-Luc Kirgizov, Sergey Ramírez, José L. Villamizar, Diego |
| contents | 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 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 $(1+x+x^2)^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06228 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Combinatorics of Motzkin Polyominoes Baril, Jean-Luc Kirgizov, Sergey Ramírez, José L. Villamizar, Diego Combinatorics Discrete Mathematics 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 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 $(1+x+x^2)^n$. |
| title | The Combinatorics of Motzkin Polyominoes |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2401.06228 |