SIP classes and four-parameter partition identities

TL;DR

This paper extends SIP classes and introduces a unified approach to study four-parameter weights of partitions, connecting classic identities with new combinatorial insights and generating functions for key partition statistics.

Contribution

It generalizes SIP classes and develops a unified method for analyzing four-parameter partition weights, linking to partition statistics like BG-rank and parity-based distributions.

Findings

Derived new four-parameter partition identities

Provided generating functions for BG-rank and parity statistics

Unified approach enhances understanding of partition weights

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.