Neighborly partitions and the numerators of Rogers-Ramanujan identities

TL;DR

This paper introduces neighborly partitions and establishes dual identities to Rogers-Ramanujan identities using a novel correspondence with monomial ideals and infinite graphs.

Contribution

It presents new partition identities and a unique framework linking partitions, algebraic ideals, and graph theory.

Findings

Proved two dual partition identities related to Rogers-Ramanujan identities.

Established a correspondence between neighborly partitions, monomial ideals, and infinite graphs.

Provided insights into the structure of partitions through algebraic and graph-theoretic methods.

Abstract

We prove two partition identities which are dual to the Rogers-Ramanujan identities. These identities are inspired by (and proved using) a correspondence between three kinds of objects: a new type of partitions (neighborly partitions), monomial ideals and some infinite graphs.