Monomial ideals induced by permutations avoiding patterns

Abstract

Let S (or T) be the set of permutations of $$[n]=\{1,\ldots ,n\}$$ avoiding 123 and 132 patterns (or avoiding 123, 132 and 213 patterns). The monomial ideals $$I_S = \langle {\mathbf {x}}^{\sigma } = \prod _{i=1}^n x_i^{\sigma (i)} : \sigma \in S \rangle $$ and $$I_T = \langle {\mathbf {x}}^{\sigma } : \sigma \in T \rangle $$ in the polynomial ring $$R = k[x_1,\ldots ,x_n]$$ over a field k have many interesting properties. The Alexander dual $$I_S^{[{\mathbf {n}}]}$$ of $$I_S$$ with respect to $${\mathbf {n}}=(n,\ldots ,n)$$ has the minimal cellular resolution supported on the order complex $$\mathbf {\Delta }(\Sigma _n)$$ of a poset $$\Sigma _n$$ . The Alexander dual $$I_T^{[{\mathbf {n}}]}$$ also has the minimal cellular resolution supported on the order complex $$\mathbf {\Delta } ({\tilde{\Sigma }}_n)$$ of a poset $${\tilde{\Sigma }}_n$$ . The number of standard monomials of the Artinian quotient $$\frac{R}{I_S^{[{\mathbf {n}}]}}$$ is given by the number of irreducible (or indecomposable) permutations of $$[n+1]$$ , while the number of standard monomials of the Artinian quotient $$\frac{R}{I_T^{[{\mathbf {n}}]}}$$ is given by the number of permutations of $$[n+1]$$ having no substring $$\{l,l+1\}$$ .