Companions to the Andrews-Gordon and Andrews-Bressoud Identities and Recent Conjectures of Capparelli, Meurman, Primc, and Primc

TL;DR

This paper derives bivariate generating functions for specific partition identities related to Andrews-Gordon and Andrews-Bressoud, connecting recent conjectures to known identities and providing bijections involving cylindric partitions.

Contribution

It introduces new generating functions for the $k=1$ cases of conjectured identities, linking them to existing identities and offering combinatorial bijections.

Findings

Derived bivariate generating functions for conjectured identities.

Established equivalence with identities of Jing, Misra, and Savage.

Provided bijections involving two-rowed cylindric partitions.

Abstract

We find bivariate generating functions for the $k=1$ cases of recently conjectured colored partition identities of Capparelli, Meurman, A. Primc, and M. Primc that are slight variants of the generating functions for the sum sides of the Andrews-Gordon and Andrews-Bressoud identities, relating them to recent work of Warnaar. This $k=1$ cases turn out to be equivalent to identities of Jing, Misra, and Savage. Finally, we provide bijections for these identities involving two-rowed cylindric partitions, in the spirit of Corteel.