Double Derangement Permutations

Abstract

Let n be a positive integer. A permutation a of the symmetric group of permutations of is called a derangement if for each . Suppose that x and y are two arbitrary permutations of . We say that a permutation a is a double derangement with respect to x and y if and for each . In this paper , we give an explicit for mula for , the number of double derangements with respect to x and y . Let and let and be two subsets of with and . Suppose that denotes the number of derangements x such that . As the main result, we show that if and z is a permutation such that for and for , then where .