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## A. Derangement 

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 

where d$_{1}$ = 0 , and ! N is subfactorial calculated by 

<img src='content_image/304037.jpg'>

<img src='content_image/304038.jpg'>

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$^{2}$: 

<img src='content_image/304042.jpg'>

$^{2}$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$_{9}$ . 

