# Article

Keywords:
monomial ideal; facet ideal; depth; Stanley depth
Summary:
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\$.
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