Signed Partitions and Rogers-Ramanujan type Identities

TL;DR

This paper develops a unified signed-partition framework to reinterpret the sum sides of Rogers-Ramanujan type identities. It provides both analytic (generating-function) and bijective proofs, yielding new signed interpretations for RR and GG identities, including a novel variant for GG beyond Andrews. It treats the two little Gollnitz identities, establishing explicit bijections between ordinary and signed partitions and deriving corresponding generating-function identities. The approach offers a natural combinatorial lens on $q$-series and suggests extensions to other identities, such as Slater's $S52$.

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 Göllnitz--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 Göllnitz--Gordon identities, different from that of Andrews. Both analytic and bijective proofs are presented.

Theorems & Definitions (37)

• Definition 1.1
• Definition 1.2
• Lemma 2.1
• proof
• Theorem 2.2
• proof : Generating function proof
• proof : Bijective Proof
• Theorem 2.3
• proof
• proof : Bijective Proof of Theorem \ref{['SD']}