Cranks for Ramanujan-type congruences of $k$-colored partitions

TL;DR

This paper develops a framework for discovering invariants called cranks that explain Ramanujan-type congruences in partition functions, using theta block theory, and applies it to colored partitions.

Contribution

It introduces a new method leveraging theta block theory to find and prove crank functions for families of partition congruences, including colored partitions.

Findings

Found a family of crank functions explaining most known congruences for colored partitions.

Utilized theta block theory to construct and analyze these crank functions.

Provided a general framework applicable to other combinatorial functions.

Abstract

Dyson famously provided combinatorial explanations for Ramanujan's partition congruences modulo $5$ and $7$ via his rank function, and postulated that an invariant explaining all of Ramanujan's congruences modulo $5$, $7$, and $11$ should exist. Garvan and Andrews-Garvan later discovered such an invariant called the crank, fulfilling Dyson's goal. Many further examples of congruences of partition functions are known in the literature. In this paper, we provide a framework for discovering and proving the existence of such invariants for families of congruences and partition functions. As a first example, we find a family of crank functions that simultaneously explains most known congruences for colored partition functions. The key insight is to utilize a powerful recent theory of theta blocks due to Gritsenko, Skoruppa, and Zagier. The method used here should be useful in the study of other combinatorial functions.