On two conjectures about pattern avoidance of cyclic permutations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913819629977600 |
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| author | Pan, Junyao |
| author_facet | Pan, Junyao |
| contents | Let $π$ be a cyclic permutation that can be expressed in its one-line form as $π= π_1π_2 \cdot\cdot\cdot π_n$ and in its standard cycle form as $π= (c_1,c_2, ..., c_n)$ where $c_1=1$. Archer et al. introduced the notion of pattern avoidance of one-line and the standard cycle form for a cyclic permutation $π$, defined as both $π_1π_2 \cdot\cdot\cdot π_n$ and its standard cycle form $c_1c_2\cdot\cdot\cdot c_{n}$ avoiding a given pattern. Let $\mathcal{A}_n(σ_1,...,σ_k; τ)$ denote the set of cyclic permutations in the symmetric group $S_n$ that avoid each pattern of $\{σ_1,...,σ_k\}$ in their one-line forms and avoid $τ$ in their standard cycle forms. In this paper, we obtain some results about the cyclic permutations avoiding patterns in both one-line and cycle forms. In particular, we resolve two conjectures of Archer et al. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_02045 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On two conjectures about pattern avoidance of cyclic permutations Pan, Junyao Combinatorics 05A05, 05A15 Let $π$ be a cyclic permutation that can be expressed in its one-line form as $π= π_1π_2 \cdot\cdot\cdot π_n$ and in its standard cycle form as $π= (c_1,c_2, ..., c_n)$ where $c_1=1$. Archer et al. introduced the notion of pattern avoidance of one-line and the standard cycle form for a cyclic permutation $π$, defined as both $π_1π_2 \cdot\cdot\cdot π_n$ and its standard cycle form $c_1c_2\cdot\cdot\cdot c_{n}$ avoiding a given pattern. Let $\mathcal{A}_n(σ_1,...,σ_k; τ)$ denote the set of cyclic permutations in the symmetric group $S_n$ that avoid each pattern of $\{σ_1,...,σ_k\}$ in their one-line forms and avoid $τ$ in their standard cycle forms. In this paper, we obtain some results about the cyclic permutations avoiding patterns in both one-line and cycle forms. In particular, we resolve two conjectures of Archer et al. |
| title | On two conjectures about pattern avoidance of cyclic permutations |
| topic | Combinatorics 05A05, 05A15 |
| url | https://arxiv.org/abs/2505.02045 |