Lattice path enumeration for semi-magic squares of size three

TL;DR

This paper develops formulas for counting directed paths in the poset of semi-magic 3x3 squares, demonstrating applications in Vandermonde convolution and symmetry formulas for Clebsch-Gordan coefficients.

Contribution

It introduces new enumeration formulas for semi-magic squares of size three and applies them to derive symmetry relations and convolution identities.

Findings

Formulas for directed path enumeration in semi-magic squares

Application to Vandermonde convolution in graded posets

Derivation of Regge symmetry formulas for Clebsch-Gordan coefficients

Abstract

We give formulas for enumerating directed paths in the graded poset of semi-magic squares of size three. We give two applications of these formulas: an advanced example of Vandermonde convolution for finite graded posets, and a direct method for deriving Regge symmetry formulas for un-normalized Clebsch-Gordan coefficients.