Integer Sequences and Monomial Ideals

Abstract

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 = \left\langle \mathbf{x}^{\sigma} = \prod_{i=1}^n x_i^{\sigma(i)} : \sigma \in W \right\rangle $ in the polynomial ring $R = k[x_1,\ldots,x_n]$ over a field $k$ 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 $\mathbf{n}=(n,\ldots,n)$ 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, \[ \dim_k\left(\frac{R}{I_W^{[\mathbf{n}]}}\right) = \sum_{r=1}^n r!~s(n,r) = {\rm Per}\left([m_{ij}]_{n \times n} \right),\] where $s(n,r)$ is the (signless) Stirling number of the first kind and ${\rm Per}([m_{ij}]_{n \times n})$ 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 $\left\lbrace \dim_k\left(\frac{R}{I_S^{[\mathbf{n}]}}\right) \right\rbrace_{n=1}^{\infty}$ are identified.