Divisibility of certain $\ell$-regular partitions by $2$

TL;DR

This paper proves new infinite families of congruences modulo 2 for certain $ ext{ell}$-regular partition functions, including self-similarity properties and lacunarity results, advancing understanding of their divisibility patterns.

Contribution

It establishes infinite families of modulo 2 congruences for $b_3(n)$ and $b_{21}(n)$, and proves lacunarity of specific series related to $b_9(n)$, addressing conjectures by Keith and Zanello.

Findings

Infinite families of congruences modulo 2 for $b_3(n)$ and $b_{21}(n)$.

Proof of self-similarities of $b_3(n)$ modulo 2.

Lacunarity of series involving $b_9(n)$ modulo powers of 2.

Abstract

For a positive integer $\ell$, let $b_{\ell}(n)$ denote the number of $\ell$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_3(n)$ and $b_{21}(n)$. We prove a specific case of a conjecture of Keith and Zanello on self-similarities of $b_3(n)$ modulo $2$. We next prove that the series $\sum_{n=0}^{\infty}b_9(2n+1)q^n$ is lacunary modulo arbitrary powers of $2$. We also prove that the series $\sum_{n=0}^{\infty}b_9(4n)q^n$ is lacunary modulo $2$.