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

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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.