An Eventown Result for Permutations

TL;DR

This paper establishes bounds on even-cycle-intersecting permutation families, confirming a conjecture and revealing structural extremal families, with implications for algebraic combinatorics and symmetry group analysis.

Contribution

It proves an upper bound on the size of even-cycle-intersecting families in symmetric groups and characterizes extremal cases, confirming a conjecture and connecting to classical combinatorial problems.

Findings

Maximum size of even-cycle-intersecting families is 2^{n-1}.

Extremal families are double-translates of Sylow 2-subgroups when n is a power of 2.

Canonical intersecting families are also extremal odd-cycle-intersecting families for even n.

Abstract

A family of permutations $\mathcal{F} \subseteq S_n$ is even-cycle-intersecting if $\sigma \pi^{-1}$ has an even cycle for all $\sigma,\pi \in \mathcal{F}$. 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 J\'anos K\"orner 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.