Developments in the Khintchine-Meinardus probabilistic method for asymptotic enumeration

Boris L. Granovsky¹ ; Dudley Stark²

¹Technion-Israel Institute of Technology
²Queen Mary, University of London

The electronic journal of combinatorics, Tome 22 (2015) no. 4

Résumé

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 Taylor 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 the above generating functions. This allows novel applications of the method. In particular, we prove rigorously the asymptotics of Gentile statistics and derive the asymptotics of combinatorial objects with distinct components.

Détail

DOI : 10.37236/4581

Classification : 05A16, 60F99, 81T25
Mots-clés : Meinardus' theorem, asymptotic enumeration, Dirichlet generating functions, models of ideal gas and of quantum field theory