Khinchin Families, Set Constructions, Partitions and Exponentials.

In this paper, we give a simple criterion to verify that functions of the form eg are in the Hayman class when g is a power series with nonnegative coefficients. Thus, using the Hayman and Báez-Duarte formulas, we obtain asymptotics for the coefficients of generating functions that arise in many examples of set construction in analytic combinatorics. This new criterion greatly simplifies the one obtained previously by the authors.