Six types of separable integer partitions

TL;DR

The article develops six types of separable integer partition classes with modulus $k$ to organize and generalize generating-function frameworks for partitions. By constructing combinatorial bases and applying $q$-series identities, it derives explicit generating functions for the primary families $\mathcal{P}_{k,r}$, $\mathcal{D}_{k,r}$ and the four parameterized sets $\mathcal{R}_{a,b,k}$, $\mathcal{R}'_{a,b,k}$, $\mathcal{D}_{a,b,k}$, and $\mathcal{D}'_{a,b,k}$. It shows these families are separable IPCs with modulus $k$ and provides closed-form product or $q$-series expressions, connecting to classical identities in partition theory (e.g., Göllnitz-Gordon). The work unifies and extends generating-function techniques for six partition types, offering explicit tools for further combinatorial and number-theoretic explorations in separable partition classes. Overall, the paper advances the structural understanding and computational toolkit for separable integer partitions under modulus constraints.

Abstract

Recently, Andrews introduced separable integer partition classes and studied some well-known theorems. In this atricle, we will investigate six types of partitions from the view of the point of separable integer partition classes.

Theorems & Definitions (22)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Lemma 3.1
• Theorem 3.2
• Lemma 4.1
• Theorem 4.2
• Lemma 4.3