Stanley depth of factors of polymatroidal ideals and the edge ideal of forests

Let \({S=\mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over the field \({\mathbb{K}}\). Suppose that \({J\subsetneq I}\) are polymatroidal ideals of S. We provide a lower bound for the Stanley depth of I/J. Using this lower bound, we prove that \({{\rm sdepth}(I^k/I^{k+1})\geq {\rm depth}(I^k/I^{k+1})}\) for every integer \({k\gg0}\). We also prove that if I is the edge ideal of a forest graph with p connected components, then \({{\rm sdepth}(I^k/I^{k+1})\geq p}\) and conclude that \({{\rm sdepth}(I^k/I^{k+1})\geq {\rm depth}(I^k/I^{k+1})}\) for every integer \({k\gg0}\).