A non-trivial intersection theorem for permutations with fixed number of cycles

Abstract

Let Sn denote the set of permutations of [n]={1,2,…,n}. For a positive integer k, define Sn,k to be the set of all permutations of [n] with exactly k disjoint cycles, i.e., Sn,k={π∈Sn:π=c1c2⋯ck}, where c1,c2,…,ck are disjoint cycles. The size of Sn,k is given by [nk]=(−1)n−ks(n,k), where s(n,k) is the Stirling number of the first kind. A family A⊆Sn,k is said to be t-cycle-intersecting if any two elements of A have at least t common cycles. A family A⊆Sn,k is said to be trivially t-cycle-intersecting if A is the stabiliser of t fixed points, i.e., A consists of all permutations in Sn,k with some t fixed cycles of length one. For 1≤j≤t, let Qj={σ∈Sn,k:σ(i)=ifor alli∈[k]∖{j}}. For t+1≤s≤k, let Bs={σ∈Sn,k:σ(i)=ifor alli∈[t]∪{s}}. In this paper, we show that, given any positive integers k,t with k≥2t+3, there exists an n0=n0(k,t), such that for all n≥n0, if A⊆Sn,k is non-trivially t-cycle-intersecting, then |A|≤|B|, where B=⋃s=t+1kBs∪⋃j=1tQj. Furthermore, equality holds if and only if A is a conjugate of B, i.e., A=β−1Bβ for some β∈Sn.