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

  • Cheng Yeaw Ku
  • Kok Bin Wong
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={πSn:π=c1c2ck},
where c1,c2,,ck are disjoint cycles. The size of Sn,k is [nk]=(1)nks(n,k), where s(n,k) is the Stirling number of the first kind. A family ASn,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 kt+1, if ASn,k is t-cycle-intersecting and nn0(k,t) where n0(k,t)=O(kt+2), then
|A|[ntkt],
with equality if and only if A is the stabiliser of t fixed points.

Published
2014-07-25
Article Number
P3.16