Signed Partitions and Rogers-Ramanujan type Identities

TL;DR

This paper explores signed partitions, extending classical identities like Rogers-Ramanujan and Göllnitz-Gordon, providing new interpretations and proofs for these identities using combinatorial and analytic methods.

Contribution

It introduces a novel signed partition interpretation of Göllnitz-Gordon identities, expanding understanding of Rogers-Ramanujan type identities.

Findings

New signed partition interpretation of Göllnitz-Gordon identities

Analytic and bijective proofs provided

Enhanced combinatorial understanding of Rogers-Ramanujan identities

Abstract

George Andrews [\emph{Bull. Amer. Math. Soc.}, 2007, 561--573] introduced the idea of a \emph{signed partiton} of an integer; similar to an ordinary integer partitions, but where some of the parts could be negative. Further, Andrews reinterpreted the classical G\"ollnitz--Gordon partition identities in terms of signed partitions. In the present work, we provide interpretations of the sum sides of Rogers--Ramanujan type identities, including a new signed partition interpretation of the G\"ollnitz--Gordon identities, different from that of Andrews. Both analytic and bijective proofs are presented.