Partitions for semi-magic squares of size three

TL;DR

This paper explores partitions of semi-magic squares of size three within the context of Clebsch-Gordan coefficients, providing new proofs, generating functions, and insights into their combinatorial structure.

Contribution

It introduces a novel approach to partitioning semi-magic squares and offers new proofs and generating functions related to Clebsch-Gordan coefficients.

Findings

Provides an alternative proof of McMahon's formula.

Develops a generating function for trivial zeros of Clebsch-Gordan coefficients.

Introduces two models for partitions based on top lines and group orbits.

Abstract

In the theory of Clebsch-Gordan coefficients, one may recognize the domain space as the set of weakly semi-magic squares of size three. Two partitions on this set are considered: a triangle-hexagon model based on top lines, and one based on the orbits under a finite group action. In addition to giving another proof of McMahon's formula, we give a generating function that counts the so-called trivial zeros of Clebsch-Gordan coefficients and its associated quasi-polynomial.