Erdos-Ko-Rado theorems for permutations and set partitions

Let Sym([n]) denote the collection of all permutations of [n] = {1,..., n}. Suppose A subset of Sym([n]) is a family of permutations such that any two of its elements (when written in its cycle decomposition) have at least t cycles in common. We prove that for sufficiently large n, vertical bar A vertical bar <= (n - t)! with equality if and only if A is the stabilizer of t fixed points. Similarly, let B(n) denote the collection of all set partitions of [n] and suppose A subset of B(n) is a family of set partitions such that any two of its elements have at least t blocks in common. It is proved that, for sufficiently large n, vertical bar A vertical bar <= Bn-t with equality if and only if A consists of all set partitions with t fixed singletons, where B-n is the nth Bell number. (C) 2008 Elsevier Inc. All rights reserved.