Where did the bases states go?

Question

I wrote out the bases states for 4 1/2-spin particles in $|S,m\rangle$ representation. I know I should have a 16-dimensional Hilbert space, but I only have 9: $$|2,2\rangle,\dots,|2,-2\rangle; |1,1\rangle,|1,0\rangle,|1,-1\rangle; |0,0\rangle$$ Where did the other 7 bases go?

Answer 1

Some values of $S$ will be repeated. For instance $S=1$ occurs three times (that makes 6 more states you did not account for) and $S=0$ occurs twice (this is the last state unaccounted for).

Since $1/2\otimes 1/2=1\oplus 0$ you can decompose $(1\oplus 0)\otimes (1\oplus 0)$ to clearly see that $S=0$ occurs twice: once from $0\otimes 0$ and once from $1\otimes 1$. Likewise the three copies of $S=1$ occur in $1\otimes 1$, $1\otimes 0$ and $0\otimes 1$ respectively.

Finally be aware that, even if you have multiple copies of the same value of $S$, the states in each copy will be different and in fact can be made orthonormal.