The Last Laugh...

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...