On the composition of an arbitrary collection of SU(2) spins: An enumerative combinatoric approach

The whole enterprise of spin compositions can be recast as simple enumerative combinatoric problems. We show here that enumerative combinatorics (Stanley 2011 Enumerative Combinatorics (Cambridge Studies in Advanced Mathematics vol 1) (Cambridge: Cambridge University Press)) is a natural setting for spin composition, and easily leads to very general analytic formulae - many of which hitherto not present in the literature. Based on it, we propose three general methods for computing spin multiplicities; namely, (1) the multi-restricted composition, (2) the generalized binomial and (3) the generating function methods. Symmetric and anti-symmetric compositions of spins are also discussed, using generating functions. Of particular importance is the observation that while the common Clebsch-Gordan decomposition - which considers the spins as distinguishable - is related to integer compositions, the symmetric and anti-symmetric compositions (where one considers the spins as indistinguishable) are obtained considering integer partitions. The integers in question here are none other than the occupation numbers of the Holstein-Primakoff bosons. The pervasiveness of q-analogues in our approach is a testament to the fundamental role they play in spin compositions. In the appendix, some new results in the power series representation of Gaussian polynomials (or q-binomial coefficients) - relevant to symmetric and antisymmetric compositions - are presented.