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Some probabilistic aspects of set partitions. (English) Zbl 0876.05005
Let \(B_n\) be the \(n\)th Bell number. The author observes that Dobiński’s formula, \[ B_n = e^{-1}\sum_{m=1}^{\infty} m^n/m!, \] implies that \(B_n\) is the \(n\)th moment of the Poisson distribution with mean 1. This leads to an exploration of the connection between set partitions and probability theory. Pitman derives formulæ for the Bell numbers, including variations of Dobiński’s formula, from probabilistic arguments and investigates the distributions of various probability functions associated with randomly distributing \(n\) balls among \(m\) boxes.

MSC:
05A18 Partitions of sets
60C05 Combinatorial probability
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