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Basis partitions and their signature

Krishnaswami Alladi

TL;DR

The paper develops a comprehensive combinatorial framework for basis partitions, linking them to Rogers–Ramanujan partitions through primary partitions and a sliding operation, and introduces the signature to refine generating functions and uncover parity phenomena via a theta-series. It provides explicit constructions and generating functions for basis partitions, prescribed-signature partitions, and complete basis partitions, and proves parity results using a Franklin-type involution. The study extends to partitions with non-repeating odd parts using 2-modular graphs, giving constructive procedures for minimal and basis partitions in that setting and revealing vector-partition interpretations and hook-length correspondences. Collectively, the work yields new combinatorial proofs and identities, connects to Lebesgue-type expansions and Basis Partition Polynomials, and broadens the landscape of partition theory with detailed structural and generating-function insights.

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.

Basis partitions and their signature

TL;DR

The paper develops a comprehensive combinatorial framework for basis partitions, linking them to Rogers–Ramanujan partitions through primary partitions and a sliding operation, and introduces the signature to refine generating functions and uncover parity phenomena via a theta-series. It provides explicit constructions and generating functions for basis partitions, prescribed-signature partitions, and complete basis partitions, and proves parity results using a Franklin-type involution. The study extends to partitions with non-repeating odd parts using 2-modular graphs, giving constructive procedures for minimal and basis partitions in that setting and revealing vector-partition interpretations and hook-length correspondences. Collectively, the work yields new combinatorial proofs and identities, connects to Lebesgue-type expansions and Basis Partition Polynomials, and broadens the landscape of partition theory with detailed structural and generating-function insights.

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.

Paper Structure

This paper contains 9 sections, 13 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. The generating function of basis partitions
  3. Constructing basis partitions from primary partitions
  4. Constructing a basis partition of a successive rank vector
  5. The signature of basis partitions and a theta series
  6. A Franklin type involution for Theorem 3
  7. Basis partitions with a prescribed signature
  8. Complete basis partitions
  9. Basis partitions among partitions with non-repeating odd parts

Key Result

Theorem 1

Let R denote the set of Rogers-Ramanujan partitions. For each partition $\pi\in$R, let its weight be $\omega(\pi)=2^t$, where t is the number of strict inequalities in (eq:2.7). Then Here $\nu(\pi)$ denotes the number of parts of a partition $\pi$.

Theorems & Definitions (13)

  • Theorem 1: Hirschhorn Hi0099
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 3 more