The 𝐴₂ Andrews–Gordon identities and 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 theA2\mathrm {A}_2(orA2(1)\mathrm {A}_2^{(1)}) analogues of the celebrated Andrews–Gordon identities. We further proveqq-series identities that correspond to the infinite-level limit of the Andrews–Gordon identities forAr−1\mathrm {A}_{r-1}(orAr−1(1)\mathrm {A}_{r-1}^{(1)}) for arbitrary rankrr. Our results forA2\mathrm {A}_2also lead to conjectural, manifestly positive, combinatorial formulas for the22-variable generating function of cylindric partitions of rank33and leveldd, such thatddis not a multiple of33.