On $P$-crucial square-free permutations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908483645865984 |
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| author | Valyuzhenich, Alexandr |
| author_facet | Valyuzhenich, Alexandr |
| contents | A permutation is square-free if it does not contain two consecutive factors of length two or more that are order-isomorphic. A square-free permutation of length $n$ is $P$-crucial, where $P$ is a subset of $\{0,1,\ldots,n\}$, if any of its extensions in any position from the set $P$ contains a square. In 2015, Gent, Kitaev, Konovalov, Linton and Nightingale initiated the study of $P$-crucial square-free permutations. In particular, they showed that $\{0,1,n-1,n\}$-crucial square-free permutations of length $n$, where $n\leq 22$, exist if and only if $n=17$ or $n=21$. In this work, we prove that for any $m\geq 2$ there exists a $\{0,1,8m+4,8m+5\}$-crucial square-free permutation of length $8m+5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06907 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $P$-crucial square-free permutations Valyuzhenich, Alexandr Combinatorics A permutation is square-free if it does not contain two consecutive factors of length two or more that are order-isomorphic. A square-free permutation of length $n$ is $P$-crucial, where $P$ is a subset of $\{0,1,\ldots,n\}$, if any of its extensions in any position from the set $P$ contains a square. In 2015, Gent, Kitaev, Konovalov, Linton and Nightingale initiated the study of $P$-crucial square-free permutations. In particular, they showed that $\{0,1,n-1,n\}$-crucial square-free permutations of length $n$, where $n\leq 22$, exist if and only if $n=17$ or $n=21$. In this work, we prove that for any $m\geq 2$ there exists a $\{0,1,8m+4,8m+5\}$-crucial square-free permutation of length $8m+5$. |
| title | On $P$-crucial square-free permutations |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.06907 |