$s$-Catalan numbers and Littlewood-Richardson polynomials

TL;DR

This paper explores two generalizations of Catalan numbers, connecting them to Littlewood-Richardson coefficients and polynomials, with origins in quantum physics and spin multiplicities.

Contribution

It provides a combinatorial description of s-Catalan numbers and spin s-Catalan numbers using Littlewood-Richardson coefficients and analyzes their properties.

Findings

Combinatorial interpretation of s-Catalan numbers

Connection to Littlewood-Richardson polynomials

Properties related to quantum physics applications

Abstract

In this note, we study two generalizations of the Catalan numbers, namely the $s$-Catalan numbers and the spin $s$-Catalan numbers. These numbers first appeared in relation to quantum physics problems about spin multiplicities. We give a combinatorial description for these numbers in terms of Littlewood-Richardson coefficients, and explain some of the properties they exhibit in terms of Littlewood-Richardson polynomials.