A Note on Stability Properties of Powers of Polymatroidal Ideals

Let I be a matroidal ideal of degree d polynomial ring $$R=K[x_1,\ldots ,x_n]$$ , where K is field. $${\text {astab}}(I)$$ and {dstab}}(I)$$ the smallest integers m n, for which {Ass}}(I^m)$$ {depth}}(I^n)$$ stabilize, respectively. In this paper, we show that {astab}}(I)=1$$ if only {dstab}}(I)=1$$ . Moreover, prove $$d=3$$ then {astab}}(I)={\text Furthermore, an almost square-free Veronese type d, {dstab}}(I)=\lceil \frac{n-1}{n-d}\rceil $$