New congruences for Andrews' singular overpartitions

Recently, Andrews defined the combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function $\overline{C}_{k, i}(n)$ which gives the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. He also proved that $\overline{C}_{3, 1}(9n + 3) \equiv \overline{C}_{3, 1}(9n + 6) \equiv 0 ({\rm mod} 3)$. Chen, Hirschhorn and Sellers then found infinite families of congruences modulo 3 and modulo powers of 2 for $\overline{C}_{3, 1}(n)$, $\overline{C}_{6, 1}(n)$ and $\overline{C}_{6, 2}(n)$. In this paper, we find new congruences for $\overline{C}_{3, 1}(n)$ modulo 4, 18 and 36, infinite families of congruences modulo 2 and 4 for $\overline{C}_{8, 2}(n)$, congruences modulo 2 and 3 for $\overline{C}_{12, 2}(n)$, $\overline{C}_{12, 4}(n)$, and congruences modulo 2 for $\overline{C}_{24, 8}(n)$ and $\overline{C}_{48, 16}(n)$. We use simple p-d...