Andrew's singular overpartitions without multiples of k

Abstract

In a recent work G. E. Andrews gave definition of combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters δ and i, can be enumerated by function C_δ,i(n), the number of overpartitions of n such that no part divisible by δ and only parts ≡ ∓ i (mod δ) may be overlined. In this paper, we establish several infinite families of congruences C^k_δ,i(n), the number of singular overpartitions of n without multiples of k such that no part divisible by δ and only parts ≡ ∓ i (mod δ) may be overlined. For example, for all n ≥ 0 and α ≥ 0, C⁵_4,1(2^2α+5n + (7 ∙ 2^α+3+1) ∕ 3) ≡ 0 (mod 2²).