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 $2$.

14 pages