A note on stability properties of powers of polymatroidal ideals

Let $I$ be a matroidal ideal of degrre $d$ of a polynomial ring $R=K[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$, where $K$ is a field. 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$(I)=1$ if and only if dstab$(I)=1$. Moreover, we prove that if $d=3$, then ${\rm astab}(I)={\rm dstab}(I)$\mathrm{.}Furthermore,weshowthatif$I$ is an almost square-free Veronese type ideal of degree $d$, then ${\rm astab}(I)={\rm dstab}(I)=\lceil\frac{n-1}{n-d}\rceil$.