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 = Indt(Pn) generated by all monomials xF, where F is an independent set of vertices of the path graph Pn of size t, which is indeed the facet ideal of the tth skeleton of the independence complex of Pn. 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.
