An analogue of the Erd{\H o}s Matching Conjecture for permutations with fixed number of cycles
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| Format: | Preprint |
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2025
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| _version_ | 1866912552053637120 |
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| author | Ku, Cheng Yeaw Wong, Kok Bin |
| author_facet | Ku, Cheng Yeaw Wong, Kok Bin |
| contents | Let $S_{n}$ denote the set of permutations of $[n]=\{1,2,\dots, n\}$. For each integer $k\geq 1$, let $S_{n,k}$ be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles.
A subset $H\subseteq S_{n,k}$ is to be a matching if $π_1$ and $π_2$ do not have any common cycles for all distinct $π_1,π_2\in H$. The matching number of a family $\mathcal A\subseteq S_{n,k}$ is denoted by $ν_{p}(\mathcal A)$ and is defined to be the size of the largest matching in $\mathcal A$. In this paper, we determine the maximum size of a family $\mathcal A\subseteq S_{n,k}$ subject to the condition $ν_p(\mathcal A)\leq s$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17683 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An analogue of the Erd{\H o}s Matching Conjecture for permutations with fixed number of cycles Ku, Cheng Yeaw Wong, Kok Bin Combinatorics Let $S_{n}$ denote the set of permutations of $[n]=\{1,2,\dots, n\}$. For each integer $k\geq 1$, let $S_{n,k}$ be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles. A subset $H\subseteq S_{n,k}$ is to be a matching if $π_1$ and $π_2$ do not have any common cycles for all distinct $π_1,π_2\in H$. The matching number of a family $\mathcal A\subseteq S_{n,k}$ is denoted by $ν_{p}(\mathcal A)$ and is defined to be the size of the largest matching in $\mathcal A$. In this paper, we determine the maximum size of a family $\mathcal A\subseteq S_{n,k}$ subject to the condition $ν_p(\mathcal A)\leq s$. |
| title | An analogue of the Erd{\H o}s Matching Conjecture for permutations with fixed number of cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.17683 |