Multisectioned Moments of Stirling Numbers of the Second Kind

The Stirling number of the second kind, S(n, k), enumerates the ways that n distinct objects can be stored in k non-empty indistinguishable boxes. When k is restricted to a given residue class modulo μ, the moments of the distribution S(n, k) have properties associated with the Olivier functions of order μ evaluated at 1 and −1. The simplest example is Dobinski's formula for the Bell number B n = ∑ k − 0 n S ( n , k ) = e − 1 ∑ λ − 0 ∞ λ n / λ ! The treatment is based on combinatorial arguments and umbral calculus.