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Congruences Modulo Powers of 7 for the Reciprocal Crank Parity Function

Dandan Chen

Abstract

Amdeberhan and Merca recently studied arithmetic properties of the sequence $a(n)$, the reciprocal of the crank parity function, which counts the number of integer partitions of weight $n$ whose even parts are monochromatic and whose odd parts may appear in one of three colors (OEIS A298311). A key result of their work was the congruence $a(7n + 2) \equiv 0 \pmod{7}$ for all $n \geq 0$. We prove new congruences for the reciprocal crank parity function modulo powers of $7$.

Congruences Modulo Powers of 7 for the Reciprocal Crank Parity Function

Abstract

Amdeberhan and Merca recently studied arithmetic properties of the sequence , the reciprocal of the crank parity function, which counts the number of integer partitions of weight whose even parts are monochromatic and whose odd parts may appear in one of three colors (OEIS A298311). A key result of their work was the congruence for all . We prove new congruences for the reciprocal crank parity function modulo powers of .

Paper Structure

This paper contains 7 sections, 7 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Modular equation
  3. A modular equation
  4. The $U_p$ Operator
  5. A Fundamental Lemma
  6. Proof of Theorem \ref{['thm-main']}
  7. The Fundamental Relations for the Reciprocal of Crank Parity Function for Powers of $7$

Key Result

Theorem 1.1

For every non-negative integer $n$, the partition function satisfies:

Theorems & Definitions (11)

  • Theorem 1.1: Ramanujan's Congruences
  • Definition 1.2: Rank
  • Definition 1.3: Crank
  • Definition 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Theorem 2.2: At-Le70
  • Lemma 2.3: Fundamental Lemma
  • Lemma 2.4: Chen-Chen-Garvan-arxiv
  • Lemma 2.5: Chen-Chen-Garvan-arxiv
  • ...and 1 more