conditions $(M_1)-(M_3)$ are satisfied by the weights (\[cknu\]) with $\rho=\nu$ and $A=C$.

Throughout the paper we assume that conditions $(M_1)-(M_3)$ are satisfied. Our aim is to determine asymptotically, as $n\to\infty$, the distribution of the maximal part size $X_n$. Recalling (\[df\]), we also point out that our results may be interpreted in terms of the asymptotic of the combinatorial counts of partitions whose part sizes are $\le m$, where the range of the values of $m$ is specified by the weak convergence of the random variable $X_n$ to a non-degenerate random variable. In the brief review given below we summarize some known results on the limiting behavior of the random variable $X_n$.

Consider first the classical case of linear integer partitions, where the weights satisfy $b_k=1, k\ge 1$. This kind of partitions were broadly studied by many authors in many respects. Their graphical representations by Ferrers diagrams show that their total number of parts and their maximal part size $X_n$ are identically distributed for all $n$ (see \[2; Sect. 1.3\]). Erdös and Lehner \[6\] were apparently the first who applied a probabilistic approach to the study of integer partitions. As a matter of fact, they found an appropriate normalization for $X_n$ 