7 A derangement is a permutation in which i is not mapped to itself [10]. While the number of distinct permutations with N stations is N ! , the number of derangements is given by a recursive formula A. Derangement d$_{N}$ = N · d$_{N}$$_{−}$$_{1}$ + ( − 1) N ≜ ! N, (13) ! N = ( N − 1)[!( N − 1)+!( N − 2)] , (14) where d$_{1}$ = 0 , and ! N is subfactorial calculated by where !0 = 1 , !1 = 0 . For example, d$_{4}$ = 9 although the number of permutations is 4! = 24 . The nine derangements are listed as follows$^{2}$: $^{2}$Note that for pairwise data exchange [2], [3], pairs of stations want to send data to each other. This corresponds to a symmetric derangement in which element ( i, j ) = 1 implies element ( j, i ) = 1 , such as P$_{1}$ , P$_{5}$ and P$_{9}$ . P$_{1}$ = P$_{2}$ = P$_{3}$ =     0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0    $^{,}$     0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0    $^{,}$     0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0    $^{,}$ P$_{4}$ = P$_{5}$ = P$_{6}$ =     0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0    $^{,}$     0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0    $^{,}$     0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0    $^{,}$ P$_{7}$ = P$_{8}$ = P$_{9}$ =     0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0    $^{,}$     0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0    $^{,}$     0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0    $^{.}$