Let ?=?[?1,…,??] be the polynomial ring in n variables over a field K with the maximal ideal ?=(?1,…,??) . Let astab(?) and dstab(?) be the smallest integer n for which Ass(??) and depth(??) stabilize, respectively. In this paper we show that astab(?)=dstab(?) in the following cases:
(i) I is a matroidal ideal and ?≤5 .
(ii) I is a polymatroidal ideal, ?=4 and ?∉Ass∞(?) , where Ass∞(?) is the stable set of associated prime ideals of I.
(iii) I is a polymatroidal ideal of degree 2.
Moreover, we give an example of a polymatroidal ideal for which astab(?)≠dstab(?) . This is a counterexample to the conjecture of Herzog and Qureshi, according to which these two numbers are the same for polymatroidal ideals.
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