= Solution
The bosonic <commutator> is important in $S_+S_-=\hbar^2n_\uparrow(n_\downarrow+1)$ and $S_-S_+=\hbar^2n_\downarrow(n_\uparrow+1)$. Therefore
$$
S^2=S_z^2+\frac{S_+S_-+S_-S_+}{2}
=\frac{\hbar^2}{4}\left[(n_\uparrow+n_\downarrow)^2+2(n_\uparrow+n_\downarrow)\right].
$$
On the $N$-particle sector,
$$
\boxed{S^2=\hbar^2\frac N2\left(\frac N2+1\right),\qquad s=\frac N2}.
$$
For example $N=0,1,2$ gives <spin> $0,\tfrac12,1$. The total <number operator> can be retained in the same identity before restricting to a sector.
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