Hall polynomials, inverse Kostka polynomials and puzzles

TL;DR

This paper introduces new combinatorial formulas for Hall polynomials and inverse Kostka polynomials, connecting them to puzzles, and enhances understanding of their structure in algebraic combinatorics.

Contribution

It provides novel combinatorial expressions for two generalizations of Littlewood--Richardson coefficients, linking them to puzzle-based models.

Findings

New combinatorial formulas for Hall polynomials

Connections established between inverse Kostka polynomials and puzzles

Enhanced understanding of polynomial structures in algebraic combinatorics

Abstract

We study two different one-parameter generalizations of Littlewood--Richardson coefficients, namely Hall polynomials and generalized inverse Kostka polynomials, and derive new combinatorial formulae for them. Our combinatorial expressions are closely related to puzzles, originally introduced by Knutson and Tao in their work on the equivariant cohomology of the Grassmannian.