  We study the Stanley depth and the Hilbert depth for $I$ and $S/I$, where
$I\subset S=K[x_1,\ldots,x_N]$ is the intersection of monomial prime ideals
with disjoint sets of variables. As an application, we obtain bounds for the
Stanley depth of $I_{n,m}^t$ and $J_{n,m}^t$, where $I_{n,m}$ is the $m$-path
ideal of the path graph of length $n$ and $J_{n,m}$ is the the $m$-path ideal
of the cycle graph of length $n$.
