Congruences for (2, 3)-regular partition with designated summands

Abstract

Let $PD_{2, 3}(n)$ count the number of partitions of $n$ with designated summands in which parts are not multiples of $2$ or $3$. In this work, we establish congruences modulo powers of 2 and 3 for $PD_{2, 3}(n)$. For example, for each \quad $n\ge0$ and $\alpha\geq0$ \quad $PD_{2, 3}(6\cdot4^{\alpha+2}n+5\cdot4^{\alpha+2})\equiv 0 \pmod{2^4}$ and $PD_{2, 3}(4\cdot3^{\alpha+3}n+10\cdot3^{\alpha+2})\equiv 0 \pmod{3}.$