Monomial Ideals Induced by Permutations Avoiding Patterns

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 \(\{l,l+1\}\).