Circular Numbers andn-set Partitions

Partitions ofn-sets and their associated Bell and Stirling numbers are well-studied combinatorial entities. Less studied is the connection between these entities and the moments of a Poisson random variable. We find a natural generalization of this connection by considering the moments of the circular random variables of ordernwhich are sums ofnindependent identically distributed Poisson random variables weighted equally about the unit circle. These are shown to have relationships to partitions of an-set whose blocks are of size divisible bym. From this a rich selection of combinatorial properties analogous to those of the striling numbers are explored. This includes Dobinski-type formulas, recurrence relations, and congruential properties. Some surprises are found in that form⩾3 the circular numbers are not unimodal and have no non-trivial recurrences in (n, k) of fixed ordernwith coefficients depending onk, yet they have many analogous congruential properties.