Some New Congruences for Andrews’ Singular Overpartition Pairs

Abstract

In a recent work, Andrews defined the combinatorial objects called singular overpartitions denoted by $\overline {C}_{k,i}(n)$ , which count the number of overpartitions of n in which no part is divisible by k and only parts congruent to ± i modulo k may be overlined. Many authors have found congruences and infinite families of congruences modulo powers of 2 and 3. In this paper, we find some new infinite families of congruences for $\overline {C}^{6}_{1,2}(n)$ modulo 27 and congruences modulo 4 for $\overline {C}^{12}_{1,5}(n)$ , $\overline {C}^{9}_{3,3}(n)$ and $\overline {C}^{15}_{5,5}(n)$ .