On the Hilbert depth of a special class of squarefree monomial ideals

Let $r$ and $n$ be two positive integers and $S=K[x_1,\dots ,x_{n+r-1}]$, the ring of polynomials in $n+r-1$ variable, over a field $K$. We consider the squarefree monomial ideal $I_{n,r}:=x_1\cdots x_{r-1}(x_r,\dots ,x_{r+n-1})\subset S$ and we prove several results regarding the Hilbert depth of $S\mathrm{/}{I}_{n,r}$. Also, we consider the special case $n=r$.