The 𝐴₂ Andrews–Gordon identities and cylindric partitions

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 A 2 \mathrm {A}_2 (or A 2 ( 1 ) \mathrm {A}_2^{(1)} ) analogues of the celebrated Andrews–Gordon identities. We further prove q q -series identities that correspond to the infinite-level limit of the Andrews–Gordon identities for A r − 1 \mathrm {A}_{r-1} (or A r − 1 ( 1 ) \mathrm {A}_{r-1}^{(1)} ) for arbitrary rank r r . Our results for A 2 \mathrm {A}_2 also lead to conjectural, manifestly positive, combinatorial formulas for the 2 2 -variable generating function of cylindric partitions of rank 3 3 and level d d , such that d d is not a multiple of 3 3 .