An Exploration of Crank Generating functions for $t$-core partitions

TL;DR

This paper introduces a family of crank generating functions that explain certain partition congruences specifically for $t$-core partitions, extending the combinatorial understanding of partition congruences.

Contribution

It presents new crank generating functions tailored for $t$-core partitions, expanding the combinatorial tools for analyzing partition congruences.

Findings

Derived crank generating functions for $t$-core partitions

Explained specific partition congruences for $t$-core partitions

Extended the combinatorial framework for partition analysis

Abstract

In 1919, Ramanujan discovered his famous congruences for the partition function. Not too long after, Freeman Dyson conjectured a combinatorial statistic existed that explained the three congruences, which he dubbed the \textit{crank}. A crank generating function for the partition function was discovered in 1988 by George Andrews and Frank Garvan. Since then other crank generating functions have been found for many other kinds of partitions. In this paper, we give a family of crank generating functions which explain some partition congruences for $t$-core partitions.