# On the composition of an arbitrary collection of $SU(2)$ spins: An Enumerative Combinatoric Approach

## TL;DR

This paper applies enumerative combinatorics to analyze the composition of $SU(2)$ spins, introducing new analytic formulas and methods for symmetric and antisymmetric cases, and highlighting the role of $q$-analogues.

## Contribution

It develops three general combinatorial methods for computing spin multiplicities and connects spin compositions with integer partitions and $q$-analogues, expanding existing literature.

## Key findings

- Introduces multi-restricted composition, generalized binomial, and generating function methods.
- Links symmetric and antisymmetric compositions to integer partitions.
- Provides new results on Gaussian polynomials and $q$-binomial coefficients.

## 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.