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

TL;DR

This paper introduces a novel graphical method for partition analysis, leading to new identities involving partitions into arithmetic progressions and sum-free Sidon sets, expanding the understanding of partition structures.

Contribution

The paper develops a new graphical representation for partitions, enabling the derivation of novel partition identities related to modular sum-free Sidon sets and arithmetic progressions.

Findings

Derived new partition identities for sets with arithmetic progressions

Established connections between sum-free Sidon sets and partition identities

Provided a graphical framework for analyzing partitions

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.