A new property of partitions with applications to the Rogers-Ramanujan identities

Abstract

The order of a partition π (relative to N) is defined as the largest i for which the number of summands in the closed interval [ i, i + N − 1] is at least i. By studying the generating function for partitions into distinct parts not exceeding 2 N with given order, we are able to derive an identity of importance in the theory of partitions.