Khinchin families, set constructions, partitions and exponentials

TL;DR

This paper introduces a simple criterion to determine when exponential functions of certain power series belong to the Hayman class, enabling easier derivation of asymptotics for combinatorial generating functions.

Contribution

The authors provide a new, simplified criterion for verifying Hayman class membership for exponential generating functions with nonnegative coefficients.

Findings

New criterion simplifies previous methods

Enables easier asymptotic analysis of combinatorial structures

Applicable to a wide range of set construction problems

Abstract

In this paper, we give a simple criterion to verify that functions of the form $e^g$ are in the Hayman class when $g$ is a power series with nonnegative coefficients. Thus, using the Hayman and B\'aez-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 that obtained previously by the authors.