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We prove that the sequences $\\{{\\rm sdepth}(S/J(G)^{(k)})\\}_{k=1}^\\infty$ and $\\{{\\rm sdepth}(J(G)^{(k)})\\}_{k=1}^\\infty$ are non-increasing and hence convergent. Suppose that $\\nu_{o}(G)$ denotes the ordered matching number of $G$. We show that for every integer $k\\geq 2\\nu_{o}(G)-1$, the modules $J(G)^{(k)}$ and $S/J(G)^{(k)}$ satisfy the Stanley's inequality. 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A. Seyed Fakhari","submitted_at":"2017-09-10T09:48:27Z","abstract_excerpt":"Let $G$ be a graph with $n$ vertices and let $S=\\mathbb{K}[x_1,\\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\\mathbb{K}$. Assume that $J(G)$ is the cover ideal of $G$ and $J(G)^{(k)}$ is its $k$-th symbolic power. We prove that the sequences $\\{{\\rm sdepth}(S/J(G)^{(k)})\\}_{k=1}^\\infty$ and $\\{{\\rm sdepth}(J(G)^{(k)})\\}_{k=1}^\\infty$ are non-increasing and hence convergent. Suppose that $\\nu_{o}(G)$ denotes the ordered matching number of $G$. We show that for every integer $k\\geq 2\\nu_{o}(G)-1$, the modules $J(G)^{(k)}$ and $S/J(G)^{(k)}$ satisfy the Stanley's inequality. 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