Arithmetic properties of 3-regular bi-partitions with designated summands

Abstract

Recently Andrews, Lewis and Lovejoy introduced the partition functions $PD(n)$ defined by the number of partitions of $n$ with designated summands and they found several modulo 3 and 4. In this paper, we find several congruences modulo 3 and 4 for $PB{D}_3(n)$, which represent the number of 3-regular bi-partitions of $n$ with designated summands. For example, for each \quad $n\ge 1$ and $\alpha \ge 0$ \quad $PB{D}_3(4\cdot 3^{\alpha +2}n+10\cdot 3^{\alpha +1})\equiv 0(\mathrm{mod}3)$.