In this paper, we introduce a family of monomial ideals with the persistence property. 23 Given positive integers n and t, we consider the monomial ideal I = Indt(Pn) generated 24 by all monomials xF , where F is an independent set of vertices of the path graph Pn of 25 size t, which is indeed the facet ideal of the tth skeleton of the independence complex of 26 Pn. We describe the set of associated primes of all powers of I explicitly. It turns out 27 that any such ideal I has the persistence property. Moreover, the index of stability of I 28 and the stable set of associated prime ideals of I are determined.
