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