Partitions with part difference conditions and Bressoudʹs conjecture

By employing Andrewsʹ generalization of Watsonʹs q-analogue of Whippleʹs theorem, Bressoud obtained an analytic identity, which specializes to most of the well-known theorems on partitions with part congruence conditions and difference conditions including the Rogers–Ramanujan identities. This led him to define two partition functions A and B depending on multiple parameters as combinatorial counterparts of his identity. Bressoud then proved that A = B for some very restricted choice of parameters and conjectured the equality to hold in full generality. We provide a proof of the conjecture for a much larger class of parameters, settling many cases of Bressoudʹs conjecture.