Set partitions and moments of random variables

Abstract

It is well known that the sequence of Bell numbers ( B n ) n ⩾ 0 ( B n being the number of partitions of the set [ n ] ) is the sequence of moments of a mean 1 Poisson random variable τ (a fact expressed in the Dobiński formula), and the shifted sequence ( B n + 1 ) n ⩾ 0 is the sequence of moments of 1 + τ . In this paper, we generalize these results by showing that both ( B n 〈 m 〉 ) n ⩾ 0 and ( B n + 1 〈 m 〉 ) n ⩾ 0 (where B n 〈 m 〉 is the number of m-partitions of [ n ] , as they are defined in the paper) are moment sequences of certain random variables. Moreover, such sequences also are sequences of falling factorial moments of related random variables. Similar results are obtained when B n 〈 m 〉 is replaced by the number of ordered m-partitions of [ n ] . In all cases, the respective random variables are constructed from sequences of independent standard Poisson processes.