Partitions with part difference conditions and Bressoud's conjecture

In the theory of partitions, the Rogers-Ramanujan identities have been the starting point for a great deal of insightful work into both analytic and combinatorial questions. Strongly motivated by the Rogers-Ramanujan identities, Isaai Schur searched for further analogous partition identities, and in 1926, he proved a mod 3 anaglogue of the Rogers-Ramanujan identities. In 1948, Henry Alder showed that if further identities exist, then more complex conditions than those stated in Schur’s theorem would be required. However, in his pathbreaking paper, by relaxing the part difference conditions of the Rogers-Ramanujan identities, Basil Gordon accomplished a remarkable partition theoretic extension of the Rogers-Ramanujan identities, which was extensively generalized by George Andrews and David Bressoud. In particular, in 1980, David Bressoud established an analytic identity, which serves as a master theorem for most of the well known partition theorems including the Rogers-Ramanujan identities. This analytic identity led him to define two partition functions as combinatorial counterparts of his identity. Bressoud then proved those functions are equal in some special cases and conjectured that this holds true in general. In my talk, I will discuss his conjecture. This talk is based on joint work with Sun Kim from Ohio State University.