Congruences and asymptotics of Andrews' singular overpartitions

Recently, singular overpartitions were defined and studied by G.E. Andrews. He showed that such partitions can be enumerated by (C) over bar (k,i)(n), the number of overpartitions of n such that no part is divisible by k and only parts. equivalent to +/- i(mod k) may be overlined. Andrews proved some congruences for (C) over bar (3,1)(n) (mod 3). The author, M.D. Hirschhorn and J.A. Sellers found infinite families of congruences for (C) over bar (3,1)(n), (C) over bar (4,1), (C) over bar (6,1)(n) and (C) over bar (6,2)(n). Z. Ahmed and N.D. Baruah obtained some new congruences for (C) over bar (3,1)(n), (C) over bar (8,2), (C) over bar (12,2)(n), (C) over bar (12,4)(n), (C) over bar (24,8)(n) and (C) over bar (48,16)(n). In this paper, we prove some new congruences for (C) over bar (3,1) (n) and (C) over bar (4,1) (n) modulo powers of 2 and congruences of (C) over bar (k,i)(n) for a family of pairs k, i. We also obtain an asymptotic formula for (C) over bar (k,i) (n) as n tends to infinity. (C) 2016 Elsevier Inc. All rights reserved.