Lyndon pairs and the lexicographically greatest perfect necklace
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912227496296448 |
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| author | Becher, Verónica Tropea, Tomás |
| author_facet | Becher, Verónica Tropea, Tomás |
| contents | Fix a finite alphabet. A necklace is a circular word. For positive integers $n$ and~$k$, a necklace is $(n,k)$-perfect if all words of length $n$ occur $k$ times but at positions with different congruence modulo $k$, for any convention of the starting position. We define the notion of a Lyndon pair and we use it to construct the lexicographically greatest $(n,k)$-perfect necklace, for any $n$ and $k$ such that $n$ divides~$k$ or $k$ divides~$n$. Our construction generalizes Fredricksen and Maiorana's construction of the lexicographically greatest de Bruijn sequence of order $n$, based on the concatenation of the Lyndon words whose length divide $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_17812 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lyndon pairs and the lexicographically greatest perfect necklace Becher, Verónica Tropea, Tomás Combinatorics Discrete Mathematics Primary 68R15, 05A05, Secondary 11K16 G.2.1 Fix a finite alphabet. A necklace is a circular word. For positive integers $n$ and~$k$, a necklace is $(n,k)$-perfect if all words of length $n$ occur $k$ times but at positions with different congruence modulo $k$, for any convention of the starting position. We define the notion of a Lyndon pair and we use it to construct the lexicographically greatest $(n,k)$-perfect necklace, for any $n$ and $k$ such that $n$ divides~$k$ or $k$ divides~$n$. Our construction generalizes Fredricksen and Maiorana's construction of the lexicographically greatest de Bruijn sequence of order $n$, based on the concatenation of the Lyndon words whose length divide $n$. |
| title | Lyndon pairs and the lexicographically greatest perfect necklace |
| topic | Combinatorics Discrete Mathematics Primary 68R15, 05A05, Secondary 11K16 G.2.1 |
| url | https://arxiv.org/abs/2405.17812 |