Overpartition pairs modulo powers of 2

An overpartition of n is a non-increasing sequence of positive integers whose sum is n in which the first occurrence of a number may be overlined. In this article, we investigate the arithmetic behavior of b(k)(n) modulo powers of 2, where b(k)(n) is the number of overpartition k-tuples of n. Using a combinatorial argument, we determine b(2)(n) modulo 8. Employing the arithmetic of quadratic forms, we deduce that b(2)(n) is almost always divisible by 2(8). Finally, with the aid of the theory of modular forms, for a fixed positive integer j, we show that b(2k)(n) is divisible by 2(j) for almost all n. (C) 2011 Elsevier B.V. All rights reserved.