Congruence Properties Modulo Powers of $2$ for $4$-Regular Partitions

Let $b_\ell(n)$ denote the number of $\ell$-regular partitions of $n$. Congruences properties modulo powers of $2$ for $b_4(n)$ were considered subsequently by Andrews-Hirschhorn-Sellers, Chen, Cui-Gu, Xia, Dai, and Ballantine-Merca. In this paper, we present an approach which can be utilized to prove the ``self-similar'' congruence property satisfied by the generating function of $b_4(n)$. As an immediate consequence, one can obtain dozens of congruence families modulo powers of $2$ enjoyed by $b_4(n)$. These results not only generalize some previous results, but also can be viewed as a supplement to Keith and Zanello's comprehensive study of the congruence properties for $\ell$-regular partition functions. Finally, we also pose several conjectures on congruence families, internal congruence families and self-similar congruence properties for $4$-, $8$- and $16$-regular partition functions.