Sequences of odd length in strict partitions I: the combinatorics of double sum Rogers-Ramanujan type identities

TL;DR

This paper explores the combinatorics of strict partitions with odd-length sequences, translating complex double sum identities into partition interpretations and providing proofs rooted in classical and modern combinatorial methods.

Contribution

It introduces a new combinatorial framework for interpreting double sum Rogers-Ramanujan type identities related to strict partitions.

Findings

Derived partition interpretations for recent double sum identities.

Provided Franklin-type involutive proofs for these identities.

Connected classical MacMahon interpretations to modern identities.

Abstract

Strict partitions are enumerated with respect to the weight, the number of parts, and the number of sequences of odd length. We write this trivariate generating function as a double sum $q$-series. Equipped with such a combinatorial set-up, we investigate a handful of double sum identities appeared in recent works of Cao-Wang, Wang-Wang, Wei-Yu-Ruan, Andrews-Uncu, Chern, and Wang, finding partition theoretical interpretations to all of these identities, and in most cases supplying Franklin-type involutive proofs. This approach dates back more than a century to P. A. MacMahon's interpretations of the celebrated Rogers-Ramanujan identities, and has been further developed by Kur\c{s}ung\"{o}z in the last decade.