"The decomposition is equivalent to the EGF

Differentiate to obtain

which implies that

by convolution of exponential generating functions and because differentiating an EGF drops the first coefficient and shifts "B""n"+1 to "z" "n"/"n"!.The EGF of the Stirling numbers of the second kind is obtained by marking every subset that goes into the partition with the term formula_31, giving

Translating to generating functions, we obtain

This EGF yields the formula for the Stirling numbers of the second kind