Multiranks and classical theta functions

TL;DR

This paper introduces multiranks and new rank and crank analogs for various partitions, using elementary methods based on classical theta functions to reveal their arithmetic properties.

Contribution

It provides novel combinatorial tools for analyzing partitions, leveraging classical theta functions in a simple, elementary approach inspired by Ramanujan and others.

Findings

New multirank and crank analogs for partitions

Elementary proofs using classical theta functions

Revealed arithmetic properties of partition types

Abstract

Multiranks and new rank/crank analogs for a variety of partitions are given, so as to imply combinatorially some arithmetic properties enjoyed by these types of partitions. Our methods are elementary relying entirely on the three classical theta functions, and are motivated by the seminal work of Ramanujan, Garvan, Hammond and Lewis.