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Circular Numbers andn-set Partitions

✍ Scribed by Daniel N. Port

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
426 KB
Volume
83
Category
Article
ISSN
0097-3165

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✦ Synopsis

Partitions of n-sets and their associated Bell and Stirling numbers are wellstudied 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 order n which are sums of n independent identically distributed Poisson random variables weighted equally about the unit circle. These are shown to have relationships to partitions of a n-set whose blocks are of size divisible by m. 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 for m 3 the circular numbers are not unimodal and have no non-trivial recurrences in (n, k) of fixed order n with coefficients depending on k, yet they have many analogous congruential properties.

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