# An S_3-symmetric Littlewood-Richardson rule

## TL;DR

This paper introduces a new 'carton rule' for calculating Littlewood-Richardson coefficients that fully explains their inherent S_3 symmetry, improving upon previous methods.

## Contribution

The paper presents a novel combinatorial rule that explicitly accounts for all six symmetries of Littlewood-Richardson coefficients, enhancing understanding and computation.

## Key findings

- The carton rule fully explains the S_3 symmetry of Littlewood-Richardson coefficients.
- Previous rules only captured part of the symmetry group.
- The new rule provides a transparent and uniform computation method.

## Abstract

The classical Littlewood-Richardson coefficients C(lambda,mu,nu) carry a natural $S_3$ symmetry via permutation of the indices. Our "carton rule" for computing these numbers transparently and uniformly explains these six symmetries; previously formulated Littlewood-Richardson rules manifest at most three of the six.