An Erdős-Ko-Rado Theorem for Permutations with Fixed Number of Cycles
Keywords:
t-intersecting family, Erdős-Ko-Rado, permutations, Stirling number of the first kind
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 [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. In this paper we show that, given any positive integers k,t with k≥t+1, if A⊆Sn,k is t-cycle-intersecting and n≥n0(k,t) where n0(k,t)=O(kt+2), then
|A|≤[n−tk−t],
with equality if and only if A is the stabiliser of t fixed points.