An Eventown Result for Permutations

TL;DR

The paper establishes a tight bound for even-cycle-intersecting families in the symmetric group: for all $n\ge 2$, any such family satisfies $|\mathcal{F}| \le 2^{n-1}$, with equality when $n=2^\ell$ and $\mathcal{F}$ is a double translate of a Sylow $2$-subgroup. It treats this as an eventown-type analogue for permutations, resolving a conjecture of Körner on maximum reversing families, and it connects to an odd-cycle-intersecting analogue with its own interesting (though secondary) results. The proof blends spectral methods for normal Cayley graphs with a novel edge-weighting derived from a folklore bijection and a new character identity involving sums of hooks and two-row characters; this yields exact eigenstructure that attains the Delsarte–Hoffman bound. Consequences include precise determinations of the Lovász theta value in the power-of-two case and links to Steinberg-like characters, suggesting broader applicability to $p$-singular families and potential generalizations beyond $p=2$. The work opens several avenues, including improving the constant in Körner’s broader conjecture, extending the framework to odd primes and other groups, and exploring permutation analogues of classic extremal problems.

Abstract

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.

Theorems & Definitions (24)

• Definition 1: The $p$-regular graph of $G$
• Theorem 2: Main Result
• Theorem 3
• Theorem 4: Delsarte--Hoffman
• Theorem 5
• Theorem 6: Murnaghan--Nakayama
• Theorem 7
• proof : Proof of Theorem \ref{['thm:hooks']}
• Theorem 8
• proof