Congruences for $\ell$-regular overpartitions and Andrews’ singular overpartitions

Abstract

Let $\overline{A}_{\ell }(n)$ be the number of overpartitions of n into parts not divisible by $\ell$. In a recent paper, Shen calls the overpartitions enumerated by the function $\overline{A}_{\ell }(n)$ as $\ell$-regular overpartitions. In this paper, we find certain congruences for $\overline{A}_{\ell }(n)$, when $\ell =4, 8$, and 9. Recently, Andrews introduced the partition function $\overline{C}_{k, i}(n)$, called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts $\equiv \pm i\pmod {k}$ may be over-lined. He also proved that $\overline{C}_{3, 1}(9n+3)$ and $\overline{C}_{3, 1}(9n+6)$ are divisible by 3. In this paper, we prove that $\overline{C}_{3, 1}(12n+11)$ is divisible by 144 which was conjectured to be true by Naika and Gireesh.