On the Hilbert depth of certain monomial ideals and applications

We study the Stanley depth and the Hilbert depth for $I$ and $S\mathrm{/}I$, where $I\subset S=K[x_1,\dots ,x_N]$ 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 $n$.