Some congruences modulo 2, 8 and 12 for Andrews' singular overpartitions

Abstract

Recently, G. E. Andrews defined combinatorial objects which he called $(k,i)$-singular overpartitions, overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv\pm i\pmod k$ may be overlined. Let the number of $(k,i)$-singular overpartitions of $n$ be\linebreak denoted by $\overline{C}_{k,i}(n)$. Andrews and Chen, Hirschhorn and Sellers noted numerous congruences modulo $2$ for $\overline{C}_{3,1}(n)$. The object of this paper is to obtain new congruences modulo $2$ for $\overline{C}_{20,5}(n)$ and modulo $8$ and $12$ for $\overline{C}_{3,1}(n)$.