Asymptotic enumeration by Khintchine-Meinardus method: Necessary and sufficient conditions for sub exponential growth

TL;DR

This paper refines the conditions under which the number of combinatorial structures with multiplicative generating functions exhibits subexponential growth, extending the understanding of their asymptotic enumeration.

Contribution

It proves the necessity of Meinardus' main condition for subexponential growth and introduces a new necessary and sufficient condition for the local limit theorem.

Findings

Established necessity of Meinardus' condition for growth rates

Derived a new necessary and sufficient condition for local limit theorem

Extended analysis to structures with weighted gaps in their supports

Abstract

In this paper we prove the necessity of the main sufficient condition of Meinardus for sub exponential rate of growth of the number of structures, having multiplicative generating functions of a general form and establish a new necessary and suffcient condition for normal local limit theorem for aforementioned structures. The latter result allows to encompass in our study structures with weights having gaps in their supports.