CYCLE INDEX OF DIRECT-PRODUCT OF PERMUTATION-GROUPS AND NUMBER OF EQUIVALENCE CLASSES OF SUBSETS OF Z-NU

Let v be a positive integer and Z(v) the residue class ring module v. Two subsets D-1 and D-2 of Z(v) are said to be equivalent if there exi st t,s is an element of Z(v) with gcd(t, v)=1 such that D-1=tD(2) + s. We are interested in the number of equivalence classes of k-subsets o f Z(v) and the number of equivalence classes of subsets of Z(v). We fi rst find the cycle index of the direct product of permutation groups, and then use it to obtain the numbers mentioned above which can be vie wed as upper bounds, respectively, for the number of inequivalent (v, k, lambda) cyclic difference sets (when k(k - 1)=lambda(v - 1)) and fo r the number of inequivalent cyclic difference sets in Z(v).