The classical partition function for an open system is given as $$ Z_{\text{max}} = \sum_{N=0}^{\infty} \dfrac{h^{-N}}{N! } \prod_{j=1}^{N} \left( \sum_{i=0}^{\infty} e^{-\beta (E_{ij}-\mu)} g_{i} \right) $$ where $E_{ij}$ is $j$th particle with in the $j$th energy state, $g_i$ is the degenecery of the $i$th energy state, $\mu$ is the chemical potential, and $N!$ is removes over-counting of indistinguishable particles.
The Bose-Einstein partition function is given as $$ Z_{\text{bos}} = \prod_{j=1}^{\infty} \left( \sum_{i=0}^{\infty} e^{-\beta (E_{j}-\mu)i} \right)^{g_{j}} $$ where $j$ is evaluates all possible energies and $g_j$ is the degeneracies of the energies.
Why is it fundamentally impossible to reduce $Z_{\text{max}}$ into $Z_{\text{bos}}$, even though both systems describe particle number as not conserved? The average number of particles of both systems is given as $$ \langle N \rangle = \dfrac{1}{\beta} \dfrac{\partial \ln(Z)}{\partial \mu } $$
