  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
$I=Ind_t(P_n)$ generated by all monomials $\textbf{x} ^F$, 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.
