Integer Sequences and Monomial Ideals

Let ${\mathfrak{S}}_n$ be the set of all permutations of $[n]=\{1,\dots ,n\}$ and let $W$ be the subset consisting of permutations $\sigma \in \mathfrak{S}n$avoiding132and312-patterns\mathrm{.}Themonomialideal$I_W = \left\langle $\mathbf{x}$^{\sigma} = \prod{i=1}^n x_i^{\sigma(i)} : \sigma \in W \right\rangle $inthepolynomialring$R = k[x_1,\ldots,x_n]$overafield$k$ is called a hypercubic ideal in the article (Certain variants of multipermutohedron ideals, Proc. Indian Acad. Sci.(Math Sci. Vol. 126, No.4, (2016), 479-500). The Alexander dual $I_W^{[\mathbf{n}]}$ of $I_W$ with respect to $\mathbf{n}=(n,\dots ,n)$ has the minimal cellular resolution supported on the first barycentric subdivision $\mathbf{B}\mathbf{d}({\mathrm{\Delta }}_{n-1})$ of an $n-1$-simplex ${\mathrm{\Delta }}_{n-1}$. We show that the number of standard monomials of the Artinian quotient $\frac{R}{I_W^{[\mathbf{n}]}}$ equals the number of rooted-labelled unimodal forests on the vertex set $[n]$. In other words, $${\dim }_k\left(\frac{R}{I_W^{[\mathbf{n}]}}\right)=\sum _{r=1}^{n}r!s(n,r)=\mathrm{P}\mathrm{e}\mathrm{r}([m_{ij}{]}_{n\times n}),$$ where $s(n,r)$ is the (signless) Stirling number of the first kind and ${\rm Per}([m_{ij}]{n \times n})$isthepermanentofthematrix$[m{ij}]$with$m_{ii}=i$and$m_{ij}=1$for$i \ne j$\mathrm{.}Forvarioussubsets$S$of$\mathfrak{S}n$ consisting of permutations avoiding patterns, the corresponding integer sequences $\left\lbrace \dim_k\left(\frac{R}{I_S^{[$\mathbf{n}$]}}\right) \right\rbrace{n=1}^{\infty}$ are identified.