A family of monomial ideals with the persistence property

In this paper we introduce a family of monomial ideals with the persistence property. Given positive integers $n$ and $t$, we consider the monomial ideal $I=In{d}_t(P_n)$ generated by all monomials ${\mathbf{\text{x}}}^F$, where $F$ is an independent set of vertices of the path graph $P_n$ of size $t$, which is indeed the facet ideal of the $t$-th skeleton of the independence complex of $P_n$. We describe the set of associated primes of all powers of $I$ explicitly. It turns out that any such ideal $I$ has the persistence property. Moreover the index of stability of $I$ and the stable set of associated prime ideals of $I$ are determined.