The Rogers-Ramanujan identities, first proved by Rogers in 1894 and rediscovered by Ramanujan a few years later, have found wide-ranging applications across various branches of mathematics. This has led to several generalizations of these two identities, such as a combinatorial generalization by Gordon (now known as the Rogers-Ramanujan-Gordon identities) and an analytic generalization by Andrews. These identities also serve as character formulas in the representation theory of infinite-dimensional Lie algebras and vertex operator algebras. In this talk, we provide an affirmative answer to Bressoud's conjecture on the combinatorial generalization of Rogers-Ramanujan identities posed in Memoirs of the American Mathematical Society (1980). An overpartition analogue of Bressoud's conjecture is also presented. This is joint work with Thomas Y. He and Alice X. H. Zhao.
