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abstract: 'We consider partitions of the positive integer $n$ whose parts satisfy the following condition: for a given sequence of non-negative numbers $\{b_k\}_{k\ge 1}$, a part of size $k$ appears in exactly $b_k$ possible types. Assuming that a weighted partition is selected uniformly at random from the set of all such partitions, we study the asymptotic behavior of the largest part $X_n$. Let $D(s)=\sum_{k=1}^\infty b_k k^{-s}, s=\sigma+iy,$ be the Dirichlet generating series of the weights $b_k$. Under certain fairly general assumptions, Meinardus (1954) has obtained the asymptotic of the total number of such partitions as $n\to\infty$. Using the Meinardus scheme of conditions, we prove that $X_n$, appropriately normalized, converges weakly to a random variable having Gumbel distribution (i.e. its distribution function equals $e^{-e^{-t}}, -\infty<t<\infty$). This limit theorem extends some known results on particular types of partitions and on the Bose-Einstein model of ideal gas.'
author:
- |
    [**Ljuben Mutafchiev**]{}\
    American University in Bulgaria, 2700 Blagoevgrad, Bulgaria\
    and Institute of Mathematics and Informatics of the\
    Bulgarian Academy of Sciences\
title: The Size of the Largest Part of Random Weighted Partitions of Large Integers
---

[**Mathematics Subject classifications:**]{} 05A17, 60C05, 60F05

Introduction and Statement of the Result
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A weighted partition 