  Let $\mathbf a=(a_1,\ldots,a_r)$ be a sequence of positive integers and
$k\geq 2$ an integer. We study $p_{k,\mathbf a}(n)$, the restricted
$k$-multipartition function associated to $\mathbf a$ and $k$. We prove new
formulas for $p_{k,\mathbf a}(n)$, its waves $W_j(n,k,\mathbf a)$'s and its
polynomial part $P_{k,\mathbf a}(n)$. Also, we give a lower bound for the
density of the set $\{n\geq 0\;:\;p_{k,\mathbf a}(n)\not\equiv 0(\bmod\;m)\}$,
where $m\geq 2$ is an integer.
