Geodesic
Depth and Stanley depth of the facet ideals of some classes of simplicial complexes
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 3, pp. 753-766
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Let $\Delta _{n,d}$ (resp.\ $\Delta _{n,d}'$) be the simplicial complex and the facet ideal $I_{n,d}=(x_{1}\cdots x_{d},x_{d-k+1}\cdots x_{2d-k},\ldots ,x_{n-d+1}\cdots x_{n})$ (resp.\ $J_{n,d}=(x_{1}\cdots x_{d},x_{d-k+1}\cdots x_{2d-k},\ldots ,x_{n-2d+2k+1}\cdots x_{n-d+2k},x_{n-d+k+1}\cdots x_{n}x_{1}\cdots x_{k})$). When $d\geq 2k+1$, we give the exact formulas to compute the depth and Stanley depth of quotient rings $S/J_{n,d}$ and $S/I_{n,d}^t$ for all $t\geq 1$. When $d=2k$, we compute the depth and Stanley depth of quotient rings $S/J_{n,d}$ and $S/I_{n,d}$, and give lower bounds for the depth and Stanley depth of quotient rings $S/I_{n,d}^t$ for all $t\geq 1$.
Let $\Delta _{n,d}$ (resp.\ $\Delta _{n,d}'$) be the simplicial complex and the facet ideal $I_{n,d}=(x_{1}\cdots x_{d},x_{d-k+1}\cdots x_{2d-k},\ldots ,x_{n-d+1}\cdots x_{n})$ (resp.\ $J_{n,d}=(x_{1}\cdots x_{d},x_{d-k+1}\cdots x_{2d-k},\ldots ,x_{n-2d+2k+1}\cdots x_{n-d+2k},x_{n-d+k+1}\cdots x_{n}x_{1}\cdots x_{k})$). When $d\geq 2k+1$, we give the exact formulas to compute the depth and Stanley depth of quotient rings $S/J_{n,d}$ and $S/I_{n,d}^t$ for all $t\geq 1$. When $d=2k$, we compute the depth and Stanley depth of quotient rings $S/J_{n,d}$ and $S/I_{n,d}$, and give lower bounds for the depth and Stanley depth of quotient rings $S/I_{n,d}^t$ for all $t\geq 1$.
DOI : 10.21136/CMJ.2017.0172-16
Classification : 13C15, 13F20, 13F55, 13P10
Keywords: monomial ideal; facet ideal; depth; Stanley depth 
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     journal = {Czechoslovak Mathematical Journal},
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     year = {2017},
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Wei, Xiaoqi; Gu, Yan. Depth and Stanley depth of the facet ideals of some classes of simplicial complexes. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 3, pp. 753-766. doi: 10.21136/CMJ.2017.0172-16

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