From boxes to polynomials: a story of generalisation

TL;DR

This paper explores a progression from simple combinatorial identities involving stacking boxes to advanced theorems involving symmetric Macdonald polynomials, illustrating a unifying framework across different mathematical generalizations.

Contribution

It introduces a hierarchical framework connecting integer, polynomial, and elliptic identities through the lens of Macdonald polynomials, revealing deep structural insights.

Findings

Identifies a unifying structure linking different levels of mathematical generalizations.

Shows how setting parameters in Macdonald polynomials transitions between identities.

Demonstrates the power of combinatorial stacking in deriving complex algebraic theorems.

Abstract

Here we will embark on a journey starting with some ostensibly inauspicious boxes. Carefully stacking them in different ways yields amazing identities. From humble beginnings at the integer version: `how many steps does it take to get from row $i$ to row $j$?' to the first upgrade: the polynomial version, before finally reaching the final upgrade: the elliptic version. Each upgrade gives a more general theorem than before. Secretly, everything is controlled by the symmetric Macdonald polynomials. Setting $q = t$ in the Macdonald polynomial takes the elliptic version of the theorem to the polynomial version. Then, letting $t$ approach $1$ reduces the polynomial version to the integer version. All the beautiful theorems and ideas come merely from stacking boxes.