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On the total number of ones associated with cranks of partitions modulo 11

Dandan Chen, Rong Chen, Siyu Yin

Abstract

In 2021, Andrews mentioned that George Beck introduced partition statistics $M_w(r,m,n)$, which denote the total number of ones in the partition of $n$ with crank congruent to $r$ modulo $m$. Recently, a number of congruences and identities involving $M_w(r,m,n)$ for some small $m$ have been developed. We establish the 11-dissection of the generating functions for $M_ω(r,11,n)-M_ω(11-r,11,n)$, where $r=1,2,3,4,5$. In particular, we discover a beautiful identity involving $M_ω(r,11,11n+6)$.

On the total number of ones associated with cranks of partitions modulo 11

Abstract

In 2021, Andrews mentioned that George Beck introduced partition statistics , which denote the total number of ones in the partition of with crank congruent to modulo . Recently, a number of congruences and identities involving for some small have been developed. We establish the 11-dissection of the generating functions for , where . In particular, we discover a beautiful identity involving .

Paper Structure

This paper contains 6 sections, 13 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Modular functions
  5. Some lemmas
  6. The proof of Theorem 1

Key Result

Theorem 1.1

For any nonnegative integer $n$, we have

Theorems & Definitions (20)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 10 more