From the post "My Very Own Partition", if such a partition is possible, that would suggest that the last energy state \(E_s last\) determines the total occupancy \(Z\). The effects of consecutive \(E_s\) cancels. This is because valid \(E_{si}\) is made the roots of \(f(E_s)\) and so the integral of \(Z_s\) must be zero between roots of \(f(E_s)\).
\(Z=\left[ \beta e^{ f(E_{ s }) } \right] ^{ E_s2 }_{ E_s1 }\)
If \({ E_s2 }\) and \({ E_s1 }\) are roots of \(f(E_s)\) then \(e^{ f(E_{ s}1)}=e^{ f(E_{ s }2)}=e^{0}=1\) and \(Z=0\)
And \(E_s last\) is the last electron shell or the highest energy state. AH AH AH...
