(\[cknu\]), they proved that $$\begin{aligned}
\label{veya}
 & & \lim_{v\to 1^-}\mu_v(\{\lambda\in\mathcal{P}: (1-v)X(\lambda)
 -\nu\mid\log{(1-v)}\mid \nonumber \\
 & & -(\nu-1)\log{\mid\log{(1-v)}\mid}
 -(\nu-1)\log{\nu}-\log{C}\le t\}) \nonumber \\
 & & =e^{-e^{-t}}, \quad -\infty<t<\infty.
 \end{aligned}$$ As it was mentioned before, the weight sequence (\[cknu\]) satisfies conditions $(M_1)-(M_3)$. Vershik and Yakubovich \[22\] studied also a more realistic model of quantum ideal gas in a $N$-dimensional space, for which the weights satisfy $\sum_{j=1}^k
 b_j=K_N k^{N/2}+O(k^{\kappa_N})$, as $k\to\infty$ ($K_N$ and $\kappa_N<N/2$ are computable constants).

The main result of this paper is obtained in terms of the uniform probability measure $\mathbb{P}=\mathbb{P}_{n,b}$ on the set $\mathcal{P}_b(n)$. Before stating it, for the sake of brevity, we introduce the following notation: $$a(n)=a(n;\rho,A)
 =\left(\frac{A\Gamma(\rho+1)\zeta(\rho+1)}{n}\right)^{\frac{1}{\rho+1}},
 \quad n\ge 1,
 \label{norm}$$ where the constants $\rho$ and $A$ are defined by condition $(M1)$.

If the weight sequence $b$, satisfies conditions $(M1)-(M3)$, then, for all real $t$, the limiting distribution of the largest part size $X_n$ is given by $$\begin{aligned}
 & & \lim_{n\to\infty} \mathbb{P}(a(n)X_n+\rho\log{a(n)}
 -(\rho-1)\log{\mid\log{a(n)}\mid}-(\rho-1)\log{\rho}-\log{A} \le
 t) \nonumber \\
 & & =e^{-e^{-t}}.\label{limth}
 \end{aligned}$$

[*Remark.*]{} One can easily compare (\[limth\]) with (\[veya\]) setting in the latter one $v=1-a(n), \nu=\rho$ and $C=A$ and observe the coinciding normalizations. We also notice that the limiting results for linear and plane partitions (see \[6,14\]) follow from 