On stability properties of powers of polymatroidal ideals

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ with the maximal ideal $\frak{m}=(x_1,...,x_n)$. Let $\astab(I)$ and $\dstab(I)$ be the smallest integer $n$ for which $\Ass(I^n)$ and $\depth(I^n)$ stabilize, respectively. In this paper we show that $\astab(I)=\dstab(I)$ in the following cases: \begin{itemize} \item[(i)] $I$ is a matroidal ideal and $n\leq 5$. \item[(ii)] $I$ is a polymatroidal ideal, $n=4$ and $\frak{m}\notin\Ass^{\infty}(I)$, where $\Ass^{\infty}(I)$ is the stable set of associated prime ideals of $I$. \item[(iii)] $I$ is a polymatroidal ideal of degree $2$. \end{itemize} Moreover, we give an example of a polymatroidal ideal for which $\astab(I)\neq\dstab(I)$. This is a counterexample to the conjecture of Herzog and Qureshi, according to which these two numbers are the same for polymatroidal ideals.