The $\mathrm{A}_2$ Andrews-Gordon identities and cylindric partitions

TL;DR

This paper extends Andrews-Gordon identities to the A2 case and their infinite-level limits, providing new q-series identities and conjectural combinatorial formulas for cylindric partitions of rank 3.

Contribution

It introduces A2 analogues of Andrews-Gordon identities and establishes q-series identities for arbitrary rank, advancing the understanding of cylindric partitions and Rogers-Ramanujan-type identities.

Findings

Derived A2 Andrews-Gordon identities.

Proved q-series identities for infinite-level limits.

Proposed conjectural formulas for cylindric partitions.

Abstract

Inspired by a number of recent papers by Corteel, Dousse, Foda, Uncu and Welsh on cylindric partitions and Rogers-Ramanujan-type identities, we obtain the $\mathrm{A}_2$ (or $\mathrm{A}_2^{(1)}$) analogues of the celebrated Andrews-Gordon identities. We further prove $q$-series identities that correspond to the infinite-level limit of the Andrews-Gordon identities for $\mathrm{A}_{r-1}$ (or $\mathrm{A}_{r-1}^{(1)}$) for arbitrary rank $r$. Our results for $\mathrm{A}_2$ also lead to conjectural, manifestly positive, combinatorial formulas for the $2$-variable generating function of cylindric partitions of rank $3$ and level $d$, such that $d$ is not a multiple of $3$.