Congruences modulo 9 for bipartitions with designated summands

Andrews, Lewis, and Lovejoy studied arithmetic properties of partitions with designated summands that are defined on ordinary partitions by tagging exactly one part among parts with equal size. A bipartition of n is an ordered pair of partitions (π1, π2) with the sum of all of the parts being n . In this paper, we investigate arithmetic properties of bipartitions with designated summands. Let PD−2(n) denote the number of bipartitions of n with designated summands. We establish several Ramanujan-like congruences and an infinite family of congruences modulo 9 satisfied by PD−2(n) .