  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.
