Separable integer partition classes with restrictions on consecutive parts

TL;DR

The paper extends the framework of separable integer partition classes to partitions with restrictions on consecutive parts and to overpartitions. It defines four modular families and proves they form separable classes with modulus $k$, providing explicit generating functions expressed through the auxiliary function $g_{k,r}(h,s)$ and $q$-binomial coefficients. The authors also develop analogous results for overpartitions, deriving detailed generating-function identities for first- and last-occurrence overlined variants and their restricted counterparts. These results unify combinatorial constructions with generating-function techniques, enabling systematic enumeration of constrained partitions and their overpartition analogues. The work expands the catalog of separable partition classes and supplies tools for deriving further partition identities under consecutive-part restrictions.

Abstract

Recently, Andrews introduced separable integer partition classes and studied some well-known theorems. In this article, we will consider the types of partitions with restrictions on consecutive parts. We will show that such partitions are separable integer partition classes and then give the generating functions for such partitions.

Theorems & Definitions (25)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 1.6
• Theorem 1.7
• Definition 2.1
• Theorem 2.2
• Corollary 2.3