Question: 60. (i) Given n pairs (XI' YI)' .. . ,(X , Y,,), let G be the group of 2 permutations of the 2 variables which

60. (i) Given n pairs (XI' YI)' .. . ,(X" , Y,,), let G be the group of 2" permutations of the 2" variables which interchange Xi and Yi in all, some, or no of the n pairs. Let Go be any subgroup of G, and let e be the number of elements in Go . Any element g E Go (except the identity) is characterized by the numbers i 1, .. . , if (r 1) of the pairs in which X i and Yi have been switched. Let

d, = Yi - Xi' and let 6(1) < . .. < 6(r-1) denote the ordered values (di1 + .. . +d,)/r corresponding to Go. Then (72) continues to hold with e - 1 in place of M. (ii) State the generalization of Problem 59 to the situation of part (i). [Hartigan (1969).]

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