Developments in the Khintchine-Meinardus probabilistic method for asymptotic enumeration

TL;DR

This paper extends Meinardus' theorem using Khintchine's method to derive asymptotics for a broader class of generating functions, enabling precise analysis of weighted partitions and combinatorial objects.

Contribution

It generalizes Meinardus' theorem to more complex generating functions involving sequences and arbitrary functions, with reformulated hypotheses for broader applicability.

Findings

Extended asymptotic formulas for weighted partitions.

Rigorous proofs for Gentile statistics asymptotics.

Analyzed combinatorial objects with distinct components.

Abstract

A theorem of Meinardus provides asymptotics of the number of weighted partitions under certain assumptions on associated ordinary and Dirichlet generating functions. The ordinary generating functions are closely related to Euler's generating function $\prod_{k=1}^\infty S(z^k)$ for partitions, where $S(z)=(1-z)^{-1}$. By applying a method due to Khintchine, we extend Meinardus' theorem to find the asymptotics of the coefficients of generating functions of the form $\prod_{k=1}^\infty S(a_kz^k)^{b_k}$ for sequences $a_k$, $b_k$ and general $S(z)$. We also reformulate the hypotheses of the theorem in terms of generating functions. This allows us to prove rigorously the asymptotics of Gentile statistics and to study the asymptotics of combinatorial objects with distinct components.