Integer Sequences and Monomial Ideals

Let $${\mathfrak {S}}_n$$ be the set of all permutations of $$[n]=\{1,\ldots ,n\}$$ and let W be the subset consisting of permutations $$\sigma \in {\mathfrak {S}}_n$$ avoiding 132 and 312-patterns. The monomial ideal $$I_W = \big \langle {\mathbf {x}}^{\sigma } = \prod _{i=1}^n x_i^{\sigma (i)} : \sigma \in W \big \rangle $$ in the polynomial ring $$R = k[x_1,\ldots ,x_n]$$ over a field k is called a hypercubic ideal in Kumar and Kumar (Proc. Indian Acad. Sci. (Math Sci.) 126(4) (2016) 479–500). The Alexander dual $$I_W^{[{\mathbf {n}}]}$$ of $$I_W$$ with respect to $${\mathbf {n}}=(n,\ldots ,n)$$ has the minimal cellular resolution supported on the first barycentric subdivision $$\mathbf {Bd}(\Delta _{n-1})$$ of an $$n-1$$ -simplex $$\Delta _{n-1}$$ . 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, $$\begin{aligned} \dim _k\left( \frac{R}{I_W^{[{\mathbf {n}}]}}\right) = \sum _{r=1}^n r!~s(n,r) = \mathrm{Per}\left( [m_{ij}]_{n \times n} \right) , \end{aligned}$$ where s(n, r) is the (signless) Stirling number of the first kind and $$\mathrm{Per}([m_{ij}]_{n \times n})$$ is the permanent of the matrix $$[m_{ij}]$$ with $$m_{ii}=i$$ and $$m_{ij}=1$$ for $$i \ne j$$ . For various subsets S of $${\mathfrak {S}}_n$$ consisting of permutations avoiding patterns, the corresponding integer sequences $$\Big \lbrace \dim _k\Big (\frac{R}{I_S^{[{\mathbf {n}}]}}\Big ) \Big \rbrace _{n=1}^{\infty }$$ are identified.