Let K be a field and S = K[x1, . . . , xn]. Let I be a monomial ideal of S and u1, . . . , ur be monomials in S. We prove that if u1, . . . , ur form a filter-regular sequence on S/I, then S/I is pretty clean if and only if S/(I, u1, . . . , ur) is pretty clean. Also, we show that if u1, . . . , ur form a filter-regular sequence on S/I, then Stanley’s conjecture is true for S/I if and only if it is true for S/(I, u1, . . . , ur). Finally, we prove that if u1, . . . , ur is a minimal set of generators for I which form either a d-sequence, proper sequence or strong s-sequence (with respect to the reverse lexicographic order), then S/I is pretty clean.
