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

TL;DR

This paper proves a conjecture by Radu and Sellers regarding congruences modulo powers of 2 for broken 3-diamond partitions using an unconventional U-sequence.

Contribution

It introduces a novel approach with an unconventional U-sequence to resolve a conjecture on broken 3-diamond partition congruences.

Findings

Confirmed the conjecture on congruences modulo powers of 2.

Provided a new method involving an unconventional U-sequence.

Enhanced understanding of parity and congruences in partition functions.

Abstract

In 2007, Andrews and Paule introduced the family of functions $\Delta_k(n)$, 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 $\Delta_3(8n+r)$ 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.