### A broken circuit ring.

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Let ${\Delta}_{n,d}$ (resp. ${\Delta}_{n,d}^{\text{'}}$) be the simplicial complex and the facet ideal ${I}_{n,d}=({x}_{1}\cdots {x}_{d},{x}_{d-k+1}\cdots {x}_{2d-k},...,{x}_{n-d+1}\cdots {x}_{n})$ (resp. ${J}_{n,d}=({x}_{1}\cdots {x}_{d},{x}_{d-k+1}\cdots {x}_{2d-k},...,{x}_{n-2d+2k+1}\cdots {x}_{n-d+2k},{x}_{n-d+k+1}\cdots {x}_{n}{x}_{1}\cdots {x}_{k})$). When $d\ge 2k+1$, 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 $t\ge 1$. 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 $t\ge 1$.

In this paper, we study the Castelnuovo-Mumford regularity of square-free monomial ideals generated in degree $3$. We define some operations on the clutters associated to such ideals and prove that the regularity is preserved under these operations. We apply these operations to introduce some classes of ideals with linear resolutions and also show that any clutter corresponding to a triangulation of the sphere does not have linear resolution while any proper subclutter of it has a linear resolution....

We define nice partitions of the multicomplex associated with a Stanley ideal. As the main result we show that if the monomial ideal $I$ is a CM Stanley ideal, then ${I}^{p}$ is a Stanley ideal as well, where ${I}^{p}$ is the polarization of $I$.