Even-Cycle-Intersecting Permutations

• Even-cycle-intersecting families are subsets of Sₙ where the difference of any two permutations always contains an even-length cycle, imposing strict combinatorial constraints.
• The extremal bound |F| ≤ 2^(n-1) is achieved when n is a power of 2, with maximal families being double-translates of a Sylow 2-subgroup.
• A spectral method using character theory and the Delsarte–Hoffman technique rigorously establishes these bounds, linking combinatorics, algebra, and graph theory.

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 (Lindzey, 18 Jan 2026).

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 (Lindzey, 18 Jan 2026).

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 (Lindzey, 18 Jan 2026).

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 (Lindzey, 18 Jan 2026).