An Eventown Result for Permutations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914263249977344 |
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| author | Lindzey, Nathan |
| author_facet | Lindzey, Nathan |
| contents | A family of permutations $\mathcal{F} \subseteq S_n$ is even-cycle-intersecting if $σπ^{-1}$ has an even cycle for all $σ,π\in \mathcal{F}$. We show that if $\mathcal{F} \subseteq S_n$ is an even-cycle-intersecting family of permutations, then $|\mathcal{F}| \leq 2^{n-1}$, and that equality holds when $n$ is a power of 2 and $\mathcal{F}$ is a double-translate of a Sylow 2-subgroup of $S_n$. This result can be seen as an analogue of the classical eventown problem for subsets and it confirms a conjecture of János Körner on maximum reversing families of the symmetric group. Along the way, we show that the canonically intersecting families of $S_n$ are also the extremal odd-cycle-intersecting families of $S_n$ for all even $n$. While the latter result has less combinatorial significance, its proof uses an interesting new character-theoretic identity that might be of independent interest in algebraic combinatorics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_12613 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Eventown Result for Permutations Lindzey, Nathan Combinatorics Discrete Mathematics Group Theory A family of permutations $\mathcal{F} \subseteq S_n$ is even-cycle-intersecting if $σπ^{-1}$ has an even cycle for all $σ,π\in \mathcal{F}$. We show that if $\mathcal{F} \subseteq S_n$ is an even-cycle-intersecting family of permutations, then $|\mathcal{F}| \leq 2^{n-1}$, and that equality holds when $n$ is a power of 2 and $\mathcal{F}$ is a double-translate of a Sylow 2-subgroup of $S_n$. This result can be seen as an analogue of the classical eventown problem for subsets and it confirms a conjecture of János Körner on maximum reversing families of the symmetric group. Along the way, we show that the canonically intersecting families of $S_n$ are also the extremal odd-cycle-intersecting families of $S_n$ for all even $n$. While the latter result has less combinatorial significance, its proof uses an interesting new character-theoretic identity that might be of independent interest in algebraic combinatorics. |
| title | An Eventown Result for Permutations |
| topic | Combinatorics Discrete Mathematics Group Theory |
| url | https://arxiv.org/abs/2601.12613 |