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The $\mathrm{A}_2$ Andrews-Gordon identities and cylindric partitions (2111.07550v3)
Published 15 Nov 2021 in math.CO, math-ph, math.MP, math.NT, and math.RT
Abstract: Inspired by a number of 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 $3$.