Linked partition ideals and Kanade--Russell conjectures

TL;DR

This paper introduces a new method for proving generating function identities for partitions within linked partition ideals, demonstrating its effectiveness through the Kanade--Russell conjectures.

Contribution

It presents a novel approach based on a conjecture by George Andrews, linking $q$-difference equations to generating functions of partitions.

Findings

Generated explicit formulas for Kanade--Russell conjectures

Validated the method's effectiveness on complex partition identities

Connected $q$-difference equations with partition generating functions

Abstract

This paper will primarily present a method of proving generating function identities for partitions from linked partition ideals. The method we introduce is built on a conjecture by George Andrews and that those generating functions satisfy some $q$-difference equations. We will come up with the generating functions of partitions in the Kanade--Russell conjectures to illustrate the effectiveness of this method.