Andrews-Gordon Type Series for Kanade-Russell Conjectures

TL;DR

This paper constructs Andrews-Gordon type positive series as generating functions for partitions satisfying specific difference conditions, providing q-series conjectures related to Kanade and Russell's combinatorial conjectures.

Contribution

It introduces new Andrews-Gordon type series for partitions, serving as generating functions for conjectured partition enumerants without claiming new identities.

Findings

Constructed positive q-series for six Kanade-Russell conjectures

Provided generating functions for missing partition counts

Formulated q-series conjectures as companions to existing conjectures

Abstract

We construct Andrews-Gordon type evidently positive series as generating functions of partitions satisfying certain difference conditions in six conjectures by Kanade and Russell. We construct generating functions for missing partition enumerants, naturally without claiming new partition identities. Thus, we obtain $q$-series conjectures as companions to Kanade and Russell's combinatorial conjectures.