In the below, I phrase the question in a physics setting (given my background). The question concerns the direct (or tensor) product of an arbitrary number of SU(2) spin-1/2 objects. The objective is to find the general multiplicities of the resulting irreducible representations.
In general, the tensor product of two spins of sizes $S_{1}$ and $S_{2}$ may be written as a direct sum of all total spin representations from $S_{max}= S_{1}+S_{2}$ up to $S_{min} = |S_{1}- S_{2}|$. That is, the total spin may assume any of the $(2 \min{S_{1}, S_{2}} +1) $ values $$S_{tot} = S_{max}, S_{max}-1,S_{max}-2, ..., S_{min}$$
Thus, for instance, for the lowest non-trivial representations of SU(2), with $S_{1}=1/2$ and $S_{2} = 1/2$, we have the well known formula $$1/2 \otimes 1/2 = 1 \oplus 0$$
Here, $\otimes$ denotes the direct product and $\oplus$ the direct sum of the corresponding irreducible total spin representations (i.e., here $S_{tot} = 1,0)$. (Physically, $S_{tot}=1$ corresponds to the "triplet" states of two spin=1/2 objects such as electrons and $S_{tot}=0$ corresponds to the "singlet" state (there are $(2 S_{tot}+1)$ states that lie in any total spin $S_{tot}$ sector).
One may apply this relation iteratively to obtain the two trivial examples
$$1/2 \otimes 1/2 \otimes 1/2 = 3/2 \oplus 1/2 \oplus 1/2$$
$$1/2 \otimes 1/2 \otimes 1/2 \otimes 1/2 = 2 \oplus 1 \oplus 1 \oplus 1 \oplus 0 \oplus 0$$
and so on.
Now here is the question. Is there a general closed form expression for the tensor product of an arbitrary number N of spin-1/2 objects? That is, what is
$$1/2 \otimes 1/2 \otimes 1/2 \otimes ... \otimes 1/2?$$
More precisely, what are the multiplicities of each of the allowed sector?
Alluding to the two examples above, for N=3 the multiplicity of the $S_{tot}=1/2$ sector was 2 while the multiplicity of the maximal spin $S_{tot} =3/2$ is unity (as it is for any N).
For N=4, the multiplicities of the $S_{tot}=1$ and 0 sectors were, respectively, 3 and 2.
What is the multiplicity of a general total spin $S_{tot}$ sector when taking the direct product of a general number N of spin-1/2 objects?