Generalizations of POD and PED partitions

TL;DR

This paper generalizes POD and PED partitions by exploring partitions with fixed residue conditions and multiplicity constraints, broadening the understanding of partition structures in number theory.

Contribution

It introduces new classes of partitions with residue-based restrictions and multiplicity conditions, extending classical partition concepts.

Findings

Defined new partition classes with residue restrictions

Analyzed properties and enumerations of these generalized partitions

Connected these concepts to existing partition theories

Abstract

Partitions with even (respectively odd) parts distinct and all other parts unrestricted are often referred to as PED (respectively POD) partitions. In this article, we generalize these notions and study sets of partitions in which parts with fixed residue(s) modulo r are distinct while all other parts are unrestricted. We also study partitions in which parts divisible by r (respectively congruent to r modulo 2r) must occur with multiplicity greater than one.