Stanley depth of factors of polymatroidal ideals and the edge ideal of forests (Q500853)

\noindent Let \(S=k[x_1,\dots,x_n]\) be the polynomial ring in \(n\) variables over a field \(k\) and \(J\subseteq I\subseteq S\) be two polymatroidal ideals. By using the affine ranks of \(I\) and \(J\), the authors provide a lower bound for the Stanley depth of \(I/J\). The affine rank is also used for proving that \(I^k/I^{k+1}\) satisfies the Stanley's inequality, that is \(\mathrm{sdepth}(I^k/I^{k+1})\geq\mathrm{depth}(I^k/I^{k+1})\), for \(k\gg0\). Moreover they show that \(I^k/I^{k+1}\), \(k\gg0\), satisfies the Stanley's inequality, when \(I\) is the edge ideals of a forest graph with \(p\) connected components.