Basis Partitions and Their Signature

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 complete. Finally, we discuss basis partitions and minimal basis partitions among partitions with non-repeating odd parts by representing them using 2-modular graphs.