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 ##IMG##
[http://ej.iop.org/images/1751-8121/51/10/105202/aaaa8faieqn003.gif] {$SU(2)$} 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 relate...