---
title: Even-Cycle-Intersecting Permutations
url: https://www.emergentmind.com/topics/even-cycle-intersecting-family-of-permutations
type: topic
---

# Even-Cycle-Intersecting Permutations

An even-cycle-intersecting family of permutations is a subset $\mathcal{F} \subseteq S_n$ with the property that for every pair $\sigma, \pi \in \mathcal{F}$, the permutation $\sigma\pi^{-1}$ has at least one cycle of even length. The study of such families in the symmetric group $S_n$ reveals deep connections with extremal combinatorics, spectral graph theory, and the representation theory of symmetric groups. The extremal problem asks for the largest possible size of such a family and for the structure of families achieving this maximum.

## 1. Definition and Structural Properties

Let $S_n$ denote the symmetric group acting on the set $[n]=\{1,2,\dots,n\}$. Every permutation $\sigma\in S_n$ decomposes as a product of disjoint cycles, where the length of a cycle is its support cardinality.

**Definition:** A family $\mathcal{F}\subseteq S_n$ is even-cycle-intersecting if for every $\sigma,\pi\in\mathcal{F}$, the permutation $\sigma\pi^{-1}$ includes at least one even-length cycle. Equivalently,
\[
\mathcal{F} \subseteq S_n \text{ is even-cycle-intersecting} \Longleftrightarrow \forall\,\sigma,\pi\in\mathcal{F}:\; \sigma\pi^{-1} \text{ has an even-length cycle.}
\]

This property imposes strong combinatorial and algebraic constraints on $\mathcal{F}$, restricting its possible structure and cardinality.

## 2. Extremal Bound and Tight Examples

The primary result states that the size of any even-cycle-intersecting family $\mathcal{F}\subseteq S_n$ satisfies the sharp bound
\[
\bigl|\mathcal{F}\bigr| \leq 2^{n-1}.
\]
When $n=2^\ell$ is a power of two, equality is achieved: the maximal families are precisely the double-translates of a Sylow 2-subgroup of $S_n$, i.e., sets of the form $\sigma P_n \tau$ for $\sigma, \tau \in S_n$, where $P_n$ is a Sylow 2-subgroup. This subgroup has order
\[
|P_n| = 2^{\sum_{i \ge 1} \lfloor n/2^i \rfloor} = 2^{n-1}, 
\]
and consists of all permutations in $S_n$ whose orders are powers of 2. Every element of $P_n$ is itself even-cycle-intersecting, and thus $P_n$ exemplifies the extremal case. For $n=2^\ell$, $P_n$ can be constructed as the automorphism group of a complete binary tree of height $\ell$, which is isomorphic to a wreath product
\[
C_2 \wr C_2 \wr \cdots \wr C_2 \quad (\ell \text{-fold wreath product}).
\]
This subgroup’s extremality is tied to the group-theoretic fact that its elements are all products of only even-length cycles or cycles of length a power of two [2601.12613].

## 3. Spectral Approach and the Delsarte–Hoffman Technique

The proof of the extremal bound proceeds via a spectral method, interpreting even-cycle-intersecting families as independent sets within the Cayley graph
\[
\Gamma_{2'}(S_n)=\mathrm{Cay}\left(S_n, G_{2'} \setminus \{ \text{id} \} \right),
\]
where $G_{2'}$ denotes the subset of $S_n$ consisting of all elements of odd order (2-regular elements). In this graph, two vertices $\sigma, \pi$ are adjacent if $\sigma\pi^{-1}$ is 2-regular; non-adjacency equates to the presence of an even-length cycle in $\sigma\pi^{-1}$. Thus, an even-cycle-intersecting family becomes an independent set in $\Gamma_{2'}(S_n)$.

To bound the independence number, a weighted adjacency matrix $A$ is constructed, satisfying:
- constant row-sum (ensuring the principal eigenvalue $\lambda_{\max}$ equals the row sum),
- non-positive entries for 2-singular differences,
- eigenvalues computable via the character theory of $S_n$.

The Delsarte–Hoffman bound applies:
\[
\alpha(\Gamma_{2'}(S_n)) \leq |S_n| \frac{-\lambda_{\min}}{\lambda_{\max}-\lambda_{\min}} = 2^{n-1},
\]
where $\lambda_{\min}$ and $\lambda_{\max}$ are the least and greatest eigenvalues of $A$. This establishes the extremal size [2601.12613].

## 4. Analogy with the Classical Eventown Problem

A direct analogy is drawn to the subset Eventown problem due to Berlekamp (answering Erdős). An Eventown family $\mathcal{A} \subseteq 2^{[n]}$ consists of all subsets of even size, where each pair meets in an even number of elements. The maximum size is $2^{\lfloor n/2 \rfloor}$, achieved by grouping the ground set into blocks of size two.

In the permutation context:
- "Subset size mod 2" is replaced by "permutation order mod 2" (presence of only even-length cycles).
- "Intersection size mod 2" is replaced by the parity of cycle lengths in $\sigma\pi^{-1}$.

Sylow 2-subgroups serve as the analog of maximal elementary abelian Eventowns, attaining the parallel power-of-two bound.

## 5. Character-Theoretic Identities and Odd-Cycle-Intersecting Families

The proof leverages new and classical character-theoretic identities. Define characters:
- $\chi^{(n-k,1^k)}$: hook-shaped irreducible character (partition with one row and $k$ single boxes),
- $\chi^{(n-k,k)}$: two-row irreducible character.

Let
\[
H_n := \sum_{k=0}^{n-1} \chi^{(n-k,1^k)}, \qquad 
B_n := \sum_{k=0}^{\lfloor n/2 \rfloor} (-1)^k \chi^{(n-k,k)}.
\]
By the Murnaghan–Nakayama rule,
\[
H_n(\sigma) = 
\begin{cases}
2^{\# \text{cycles}(\sigma)-1} & \text{if all cycles of } \sigma \text{ are odd,} \\
0 & \text{otherwise,}
\end{cases}
\]
and for even $n$,
\[
B_n(\sigma) = 
\begin{cases}
2^{\# \text{cycles}(\sigma)} & \text{if all cycles of } \sigma \text{ are even,} \\
0 & \text{otherwise.}
\end{cases}
\]
The identity for $H_n$ recovers Regev's result, while the formula for $B_n$ is new and crucial for bounding the sizes of odd-cycle-intersecting families. The weighted adjacency matrices built using these character sums yield eigenvalues required for the Delsarte–Hoffman bound [2601.12613].

## 6. Related Problems, Extensions, and Open Questions

The established extremal bound confirms a conjecture of János Körner on reversing families, showing that any such family in $S_n$ has size at most $2^n$. This result invites generalizations:
- For other primes $p$, analogous $p$-singular intersection bounds in various finite groups are of interest. Typically, the Steinberg character fulfills the role of $H_n$ or $B_n$.
- It remains unresolved whether, for non-power-of-2 values of $n$, the only maximal even-cycle-intersecting families are double-translates of some Sylow 2-subgroup. No counterexamples are known.
- Further combinatorial phenomena analogous to the "Oddtown" problem (imposing oddness constraints on single cycles and differences) are suggested as directions for future investigation.

The synthesis of combinatorial and representation-theoretic arguments in this area yields an exact extremal bound and a complete characterization of equality when $n$ is a power of 2, unifying perspectives from graph theory, algebra, and extremal set theory [2601.12613].

Source: https://www.emergentmind.com/topics/even-cycle-intersecting-family-of-permutations