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A conjecture of Radu and Sellers on congruences modulo powers of 2 for broken 3-diamond partitions

Dandan Chen, Rong Chen, Siyu Yin

Abstract

In 2007, Andrews and Paule introduced the family of functions $Δ_k(n)$, which enumerate the number of broken $k$-diamond partitions for a fixed positive integer $k$. In 2013, Radu and Sellers completely characterized the parity of $Δ_3(8n+r)$ for certain values of $r$ and proposed a conjecture on congruences modulo powers of $2$ for broken $3$-diamond partitions. In this paper, we employ an unconventional $U$-sequence to resolve the revised conjecture put forward by Radu and Sellers.

A conjecture of Radu and Sellers on congruences modulo powers of 2 for broken 3-diamond partitions

Abstract

In 2007, Andrews and Paule introduced the family of functions , which enumerate the number of broken -diamond partitions for a fixed positive integer . In 2013, Radu and Sellers completely characterized the parity of for certain values of and proposed a conjecture on congruences modulo powers of for broken -diamond partitions. In this paper, we employ an unconventional -sequence to resolve the revised conjecture put forward by Radu and Sellers.

Paper Structure

This paper contains 4 sections, 11 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modular equation
  4. Radu and Sellers' conjecture

Key Result

Lemma 2.4

For $\alpha \in \mathbb{N}^{\ast}$, we have $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (26)

  • Conjecture 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 16 more