Copartitions

TL;DR

This paper introduces the theory of copartitions, a generalization of partitions, connecting to various classical topics in partition theory and providing new generating functions through analytic and combinatorial methods.

Contribution

It develops the foundational theory of copartitions and presents novel three-parameter generating functions with potential broad implications in partition theory.

Findings

Derived two forms of the three-parameter generating function

Connected copartitions to Rogers-Ramanujan and mock theta functions

Explored special cases demonstrating broader impact

Abstract

We develop the theory of copartitions, which are a generalization of partitions with connections to many classical topics in partition theory, including Rogers-Ramanujan partitions, theta functions, mock theta functions, partitions with parts separated by parity, and crank statistics. Using both analytic and combinatorial methods, we give two forms of the three-parameter generating function, and we study several special cases that demonstrate the potential broader impact the study of copartitions may have.