closely related to some recent results on the maximal particle energy in the Bose-Einstein model of ideal gas. The general setting and the probabilistic frame of problems from statistical mechanics and their relationship with enumerative combinatorics were given by Vershik \[21\]. In the context of the infinite product formula (\[euler\]), he studied the Bose-Einstein model by a family of probability measures $\mu_v,
v\in (0,1)$, defined on the set of all $b$-weighted partitions $\mathcal{P}_b =\cup_{n\ge 0}\mathcal{P}_b(n)$. So, for a partition $\lambda=(\lambda_1,...,\lambda_l)\in\mathcal{P},
\lambda_1\ge...\ge\lambda_l>0$, let $r_k(\lambda)=\{j:\lambda_j=k\}$ denote the number of parts of $\lambda$ that are equal to $k\ge 1$. Then $\mu_v$ is defined by $$\mu_v(\{\lambda\in\mathcal{P}: r_k(\lambda)=j\})={b_k+j-1\choose
j}
 v^{kj}(1-v^k)^{b_k}, \quad 0<v<1.$$ The key feature in the study of this kind of measures is the fact that a kind of a conditional probability measure on $\mathcal{P}_b(n)$ turns out to be independent of $v$ for all $n$ and coincides with the uniform probability measure $\mathbb{P}=\mathbb{P}_{n,b}$ (for more details, see \[21\]). In \[22\] Vershik and Yakubovich studied the limiting distribution of the maximal particle energy $X(\lambda)$, or, which is the same, the largest part size $X(\lambda), \lambda\in\mathcal{P}$, with respect to the measure $\mu_v$, as $v\to
 1^-$. In particular, under the assumption that the weight sequence $b$ satisfies 