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SIP classes and four-parameter partition identities

Runqiao Li

Abstract

The four-parameter weight of partitions played an important role in the theory of integer partitions, for its connection with various statistics, including the alternating sum and the BG-rank. In 2022, Andrews introduced the SIP classes, by which he reviewed a number of classic partition identities and provided new combinatorial insights. In this work, we extend the SIP classes and provide a unified method to study the four-parameter weight of partitions. By treating partitions with position parity as examples, we provide four-parameter partition identities related to these partition sets. And as corollary, we also present the generating functions that keep track of the BG-rank and the joint distribution of the number of odd parts and the alternating sum, respectively.

SIP classes and four-parameter partition identities

Abstract

The four-parameter weight of partitions played an important role in the theory of integer partitions, for its connection with various statistics, including the alternating sum and the BG-rank. In 2022, Andrews introduced the SIP classes, by which he reviewed a number of classic partition identities and provided new combinatorial insights. In this work, we extend the SIP classes and provide a unified method to study the four-parameter weight of partitions. By treating partitions with position parity as examples, we provide four-parameter partition identities related to these partition sets. And as corollary, we also present the generating functions that keep track of the BG-rank and the joint distribution of the number of odd parts and the alternating sum, respectively.
Paper Structure (9 sections, 29 theorems, 92 equations)

This paper contains 9 sections, 29 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Separable integer partition classes
  3. Strict partitions with position parity
  4. Strict partitions with even indexed parts being even
  5. Strict partitions with odd-indexed parts being even
  6. Ordinary partitions with position parity
  7. Ordinary partitions with even-indexed parts being even
  8. Ordinary partitions with odd-indexed parts being even
  9. Conclusion

Key Result

Theorem 1.1

Theorems & Definitions (47)

  • Theorem 1.1: Savage and Sills, 2011
  • Theorem 1.2: Li, 2025
  • Theorem 1.3: Andrews, 2004
  • Remark 1.4
  • Theorem 1.5: Boulet, 2006
  • Theorem 1.6
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: Andrews, 2022
  • Theorem 2.4
  • ...and 37 more