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

Abstract: 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 ${a_1,\ldots,a_k}$. We examine several arithmetic properties of the sequence $(p_\mathcal{A}(n,k) \pmod{m}){n\in\mathbb{N}}$ for an arbitrary fixed integer $m\geqslant2$. We investigate periodicity of the sequence and lower and upper bounds for the density of the set ${n\in\mathbb{N}: p\mathcal{A}(n,k) \equiv i \pmod{m}}$ for a fixed positive integer $k$ and $i\in{0,1,\ldots, m-1}$. 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.