Basis partitions and their signature

TL;DR

This paper explores the combinatorial structure of basis partitions, introduces a signature concept, and examines special classes and representations of these partitions, connecting them to Rogers-Ramanujan partitions and modular graphs.

Contribution

It provides a new combinatorial framework for basis partitions, defines their signatures, and analyzes their properties and classifications, including connections to Rogers-Ramanujan partitions.

Findings

Basis partitions can be generated from primary partitions.

A signature for basis partitions explains certain parity results.

Complete basis partitions are characterized and studied.

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.