Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 046, 38 pages      arXiv:2306.16251      https://doi.org/10.3842/SIGMA.2026.046
Contribution to the Special Issue on Recent Advances in Vertex Operator Algebras in honor of James Lepowsky

Companions to the Andrews-Gordon and Andrews-Bressoud Identities and Recent Conjectures of Capparelli, Meurman, Primc, and Primc

Matthew C. Russell
Department of Statistics, Texas A&M University, USA

Received September 05, 2025, in final form April 28, 2026; Published online May 12, 2026

Abstract
We find bivariate generating functions for the $k=1$ cases of recently conjectured colored partition identities of Capparelli, Meurman, A. Primc, and M. Primc that are slight variants of the generating functions for the sum sides of the Andrews-Gordon and Andrews-Bressoud identities, relating them to recent work of Warnaar. This $k=1$ cases turn out to be equivalent to identities of Jing, Misra, and Savage. Finally, we provide bijections for these identities involving two-rowed cylindric partitions, in the spirit of Corteel.

Key words: integer partitions; partition identities; $q$-series identities.

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