in this case and showed that $\pi X_n/(6n)^{1/2}-\log{((6n)^{1/2}/\pi)}$ converges weakly, as $n\to\infty$, to a random variable having the extreme value (Gumbel) distribution. The local version of their theorem was derived later by Auluck et al. \[4\]. Fristedt \[9\] studied linear integer partitions using a transfer method to functionals of independent and geometrically distributed random variables. Among other results, he obtained the limiting distribution of the $k$th largest part size whenever $k$ is fixed. Finally, we notice that among weighted integer partitions only the linear ones possess the property that the number of parts and the maximum part size are identically distributed. The limiting distribution of the number of parts in the general case of random weighted partitions under the Meinardus scheme of conditions is studied in \[15\]. It turns out that the limiting distribution laws depend on particular ranges in which the parameter $\rho$ varies (see condition $(M_1)$).

Another important particular case of weighted partitions arises whenever $b_k=k, k\ge 1$. It turns out that in this case the generating function $f_b(x)$ (see (\[euler\])) enumerates the plane partitions. A plane partition of $n\ge 1$ is a matrix of non-negative integers arranged in non-increasing order from left to right and from top 