SIGMA 21 (2025), 104, 17 pages      arXiv:2507.14734      https://doi.org/10.3842/SIGMA.2025.104
Contribution to the Special Issue on Basic Hypergeometric Series Associated with Root Systems and Applications in honor of Stephen C. Milne

Basis Partitions and Their Signature

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

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

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.

Key words: basis partitions; Rogers-Ramanujan partitions; Durfee squares; sliding operation; signature; partial theta series.

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