Basis Partitions and Their Signature

Krishnaswami Alladi

Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, USA

Abstract: Basis partitions are minimal partitions corresponding to successive rank vectors. We show combinatorially how basis partitions can be generated from primary partitions which are equivalent to the Rogers–Ramanujan partitions. This leads to the definition of a signature of a basis partition that we use to explain certain parity results. We then study a special class of basis partitions which we term as complete. Finally, we discuss basis partitions and minimal basis partitions among partitions with non-repeating odd parts by representing them using 2-modular graphs.

Keywords: basis partitions, Rogers–Ramanujan partitions, Durfee squares, sliding operation, signature, partial theta series.

MSC: 05A17, 05A19, 05A15

Received: July 23, 2025; in final form November 28, 2025; Published online December 11, 2025

Language: English

DOI: 10.3842/SIGMA.2025.104

ArXiv: 2507.14734