The Number of Cycles Associated with the Elements of a Permutation Group

Gontcharoff 1 has given several moment generating functions for the number of cycles associated with the elements of the permutation group on N symbols. Recently the author rediscovered the factorial moment generating function for the above problem previously given by Gontcharoff. Since the method of derivation is quite different from that given by Gontcharoff, it may be worth reproducing here. For each positive integer N, we let GN denote the group of N! permutations on N symbols (which we may refer to as being cards numbered from 1 to N for convenience). As is well known, each of these permutations may be written as a product of disjoint cycles of lengths ti, t2, * , t. say, and these lengths will satisfy the relation Eti =N. We call the permutation (1, 2, * , N) the standard permutation. The factorial moment generating function found by the author is based on the form