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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2409.17482 |
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| _version_ | 1866912047130738688 |
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| author | Pan, Junyao |
| author_facet | Pan, Junyao |
| contents | Let $π$ be a cycle permutation that can be expressed as one-line $π= π_1π_2 \cdot\cdot\cdot π_n$ and a cycle form $π= (c_1,c_2, ..., c_n)$. Archer et al. introduced the notion of pattern avoidance of one-line and all cycle forms for a cycle permutation $π$, defined as $π_1π_2 \cdot\cdot\cdot π_n$ and its arbitrary cycle form $c_ic_{i+1}\cdot\cdot\cdot c_nc_1c_2\cdot\cdot\cdot c_{i-1}$ avoid a given pattern. Let $\mathcal{A}^\circ_n(σ; τ)$ denote the set of cyclic permutations in the symmetric group $S_n$ that avoid $σ$ in their one-line form and avoid $τ$ in their all cycle forms. In this note, we prove that $|\mathcal{A}^\circ_n(2431; 1324)|$ is the $(n-1)^{\rm{st}}$ Pell number for any positive integer $n$. Thereby, we give a positive answer to a conjecture of Archer et al. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17482 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a conjecture about pattern avoidance of cycle permutations Pan, Junyao Combinatorics Let $π$ be a cycle permutation that can be expressed as one-line $π= π_1π_2 \cdot\cdot\cdot π_n$ and a cycle form $π= (c_1,c_2, ..., c_n)$. Archer et al. introduced the notion of pattern avoidance of one-line and all cycle forms for a cycle permutation $π$, defined as $π_1π_2 \cdot\cdot\cdot π_n$ and its arbitrary cycle form $c_ic_{i+1}\cdot\cdot\cdot c_nc_1c_2\cdot\cdot\cdot c_{i-1}$ avoid a given pattern. Let $\mathcal{A}^\circ_n(σ; τ)$ denote the set of cyclic permutations in the symmetric group $S_n$ that avoid $σ$ in their one-line form and avoid $τ$ in their all cycle forms. In this note, we prove that $|\mathcal{A}^\circ_n(2431; 1324)|$ is the $(n-1)^{\rm{st}}$ Pell number for any positive integer $n$. Thereby, we give a positive answer to a conjecture of Archer et al. |
| title | On a conjecture about pattern avoidance of cycle permutations |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.17482 |