An Erdős-Ko-Rado Theorem for Permutations with Fixed Number of Cycles

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 [ n k ] = (−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| 6 [ n− t k − t ] , with equality if and only if A is the stabiliser of t fixed points.