of the positive integer $n$ is a multiset of size $n$ whose decomposition into a union of disjoint components (parts) satisfies the following condition: for a given sequence of non-negative numbers $\{b_k\}_{k\ge 1}$, a part of size $k$ appears in exactly one of $b_k$ possible types. For more details on properties of multisets, we refer the reader e.g. to \[3\]. Weighted partitions are also associated with the generalized Bose-Einstein model of ideal gas, where $n(=E)$ is interpreted as the total energy of the system of particles. The weights $b_k,
k\ge 1$, are viewed as counts of the distinct positions of the particles in the state space, where a particle in a given position has (rescaled) energy $k$ (for more details on the relationship between combinatorial partitions and various models of ideal gas, see \[21\]). From combinatorial point of view, it is fairly natural to assume that $b_k, k\ge 1$, are integers (see e.g. the“money changing problem” discussed in detail in \[24; Sect. 3.15\]). On the other hand, it turns out that this requirement is not necessary for the analytical approach used in this paper. That is why, we assume that $b_k, k\ge 1$, are real non-negative numbers.

For a given 