Title: On the composition of an arbitrary collection of $SU(2)$ spins: An Enumerative Combinatoric Approach,
Abstract: The whole enterprise of spin compositions can be recast as simple enumerative
combinatoric problems. We show here that enumerative combinatorics
(EC)\citep{book:Stanley-2011} 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 $SU(2)$ spins are also discussed, using
generating functions. Of particular importance is the observation that while
the common Clebsch-Gordan decomposition (CGD) -- 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 but the occupation numbers of the
Holstein-Primakoff bosons.
\par 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.