Double Derangement Permutations

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 formula 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 .