Abstract
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 \({k\gg0}\).
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Siamak Yassemi was in part supported by a grant from University of Tehran (No. 6103023/1/014).
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Alipour, A., Seyed Fakhari, S.A. & Yassemi, S. Stanley depth of factors of polymatroidal ideals and the edge ideal of forests. Arch. Math. 105, 323–332 (2015). https://doi.org/10.1007/s00013-015-0809-7
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DOI: https://doi.org/10.1007/s00013-015-0809-7