enumerating group actions of finite cyclic groups on finite sets

This paper provides a systematic enumeration of actions of a finite cyclic group z/nz on a finite set of size k. we prove that the total number of such actions equals the number of permutations in the symmetric group sk whose order divides n, and we give an explicit combinatorial sum over cycle type vectors. furthermore, we show that the number of isomorphism classes of such actions equals the number pd(n)(k) of integer partitions of k into parts drawn from the divisor set d(n) of n. the results connect elementary number theory, partition theory, and permutation group combinatorics. we also briefly discuss extensions to free groups and elementary abelian p-groups, identifying obstacles.