Dyson's new symmetry and generalized Rogers-Ramanujan identities

Abstract

We present a generalization, which we call (k,m)-rank, of Dyson's notion of rank to integer partitions with k successive Durfee rectangles and give two combinatorial symmetries associated with this new definition. We prove these symmetries bijectively. Using the two symmetries we give a new combinatorial proof of generalized Roger-Ramanujan identities. We also describe the relationship between (k,m)-rank and Garvan's k-rank.