N-graphs, modular Sidon and sum-free sets, and partition identities

Abstract

Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of a set that is a finite union of arithmetic progressions associated with a modular sum-free Sidon set. Partition identities are also constructed for sets associated with modular sum-free sets.