A note on the restricted partition function $p_{\mathcal{A}}(n,k)$

Let $\mathcal{A}=(a_n{)}_{n\in {\mathbb{N}}_{+}}$ be a sequence of positive integers. Let $p_{\mathcal{A}}(n,k)$ denote the number of multi-color partitions of $n$ into parts in $\{a_1,\dots ,a_k\}$. We examine several arithmetic properties of the sequence $(p_{\mathcal{A}}(n,k)(\mathrm{m}\mathrm{o}\mathrm{d}m){)}_{n\in \mathbb{N}}$ for an arbitrary fixed integer $m⩾2$. We investigate periodicity of the sequence and lower and upper bounds for the density of the set $\{n\in \mathbb{N}:p_{\mathcal{A}}(n,k)\equiv i(\mathrm{m}\mathrm{o}\mathrm{d}m)\}$ for a fixed positive integer $k$ and $i\in \{0,1,\dots ,m-1\}$. In particular, we apply our results to the special cases of the sequence $\mathcal{A}$. Furthermore, we present some results related to restricted $m$-ary partitions.