On some infinite families of congruences for [j, k]-partitions into even parts distinct

Abstract

In this paper, we define the partition function $ped_{j, k}(n),$ the number of [j, k]-partitions of n into even parts distinct, where none of the parts are congruent to $j \;{\text{(mod k)}}$ (where $k >j\ge 1)$. We obtain many infinite families of congruences modulo powers of 2 for $ped_{3, 6}(n)$ and congruences modulo powers of 2 and 3 for $ped_{9, 18}(n)$. For example, for all $n \ge 0$ and $\alpha , \beta \ge 0,$

$$\begin{aligned} ped_{9, 18}\left(2\cdot 3^{4\alpha +4}\cdot 7^{2\beta +1} (7n+s)+\dfrac{11\cdot 3^{4\alpha +3}\cdot 7^{2\beta +1}+1}{4}\right)\equiv 0 \;{\text{(mod 16)}}, \end{aligned}$$

where $s = 0, 2, 3, 4, 5, 6.$