Arithmetic properties for generalized cubic partitions and overpartitions modulo a prime

TL;DR

This paper introduces generalized cubic partitions and overpartitions, establishing new Ramanujan-type congruences using elementary and modular form methods, expanding the understanding of partition arithmetic properties.

Contribution

It presents the concept of generalized cubic and overcubic partitions and proves new congruences using two distinct proof techniques.

Findings

New Ramanujan-type congruences for generalized cubic partitions

Elementary proof methods based on functional equations

Modular form-based proofs for overcubic partitions

Abstract

A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We emphasize two methods of proofs, one elementary (relying significantly on functional equations) and the other based on modular forms. We close by proving analogous results for generalized overcubic partitions.