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A Weighted Words Study of MacMahon's and Russell's Modulo 6 Identities

Ali K. Uncu

Abstract

We give new proofs of MacMahon and Russell's modulo 6 identities using the method of weighted words. We also present a new refinement of MacMahon's identity, some related finite sum identities, and a companion partition theorem to sequence avoiding partitions theorem of the author and Andrews.

A Weighted Words Study of MacMahon's and Russell's Modulo 6 Identities

Abstract

We give new proofs of MacMahon and Russell's modulo 6 identities using the method of weighted words. We also present a new refinement of MacMahon's identity, some related finite sum identities, and a companion partition theorem to sequence avoiding partitions theorem of the author and Andrews.
Paper Structure (8 sections, 14 theorems, 49 equations)

This paper contains 8 sections, 14 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Necessary Definitions and Some Background
  3. A proof of MacMahon's theorem by 2-color weighted words
  4. A proof of Russell's Theorem by 3-color weighted words
  5. Russell's Refinement of MacMahon's Theorem
  6. Some Sum Representations and Connections with Sequences in Overpartitions
  7. Outlook
  8. Acknowledgement

Key Result

Theorem 1.1

Let $n$ be a non-negative integer. The number of partitions of $n$ with no consecutive integers as parts and all parts $\geq 2$ is equal to the number of partitions of $n$, where no part occurs exactly once, is equal to the number of partitions of $n$ into parts congruent to $0,2,3$ or $4$ modulo 6.

Theorems & Definitions (14)

  • Theorem 1.1: MacMahon
  • Theorem 1.2: Russell
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 5.1
  • Theorem 5.2
  • ...and 4 more