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

Samuel Wilson

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.

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

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 -core partitions.
Paper Structure (5 sections, 4 theorems, 35 equations, 1 table)

This paper contains 5 sections, 4 theorems, 35 equations, 1 table.

Table of Contents

  1. Introduction
  2. Radu's Lemma
  3. Main Theorems
  4. Conclusion and Discussion
  5. Statement of Funding, Conflicts of Interest, and Data Availability

Key Result

Theorem 1.2

Let $p^{(5)}(n)$ denote the $5$-core partition function. Then, is a crank generating function for $p^{(5)}(n)$ which explains the congruences $p^{(5)}(15n+\beta) \equiv 0 \pmod{3}$, where $\beta \in \{6,10,12,13\}$.

Theorems & Definitions (9)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof : Proof of Theorem \ref{['thm:5crank']}
  • proof : Proof of Theorem \ref{['thm:multcrank']}