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 $PBD_{3}(n)$, which represent the number of 3-regular bi-partitions of $n$ with designated summands. For example, for each \quad $n\ge1$ and $\alpha\geq0$ \quad $PBD_{3}(4\cdot3^{\alpha+2}n+10\cdot3^{\alpha+1})\equiv 0 \pmod{3}$.