Abstract

Let v be a positive integer and Z v the residue class ring modulo v. Two subsets D 1 and D 2 of Z v are said to be equivalent if there exist t, sϵ 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 of Z v and the number of equivalence classes of subsets of Z v . We first find the cycle index of the direct product of permutation groups, and then use it to obtain the numbers mentioned above which can be viewed as upper bounds, respectively, for the number of inequivalent ( v, k, λ) cyclic difference sets (when k( k −1)= λ( v−1)) and for the number of inequivalent cyclic difference sets in Z v .

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