Monomial ideals induced by permutations avoiding patterns

Abstract

Let S (or T) be the set of permutations of [𝑛]={1,…,𝑛} avoiding 123 and 132 patterns (or avoiding 123, 132 and 213 patterns). The monomial ideals 𝐼𝑆=⟨𝐱𝜎=∏𝑛𝑖=1𝑥𝜎(𝑖)𝑖:𝜎∈𝑆⟩ and 𝐼𝑇=⟨𝐱𝜎:𝜎∈𝑇⟩ in the polynomial ring 𝑅=𝑘[𝑥1,…,𝑥𝑛] over a field k have many interesting properties. The Alexander dual 𝐼[𝐧]𝑆 of 𝐼𝑆 with respect to 𝐧=(𝑛,…,𝑛) has the minimal cellular resolution supported on the order complex 𝚫(Σ𝑛) of a poset Σ𝑛. The Alexander dual 𝐼[𝐧]𝑇 also has the minimal cellular resolution supported on the order complex 𝚫(Σ̃ 𝑛) of a poset Σ̃ 𝑛. The number of standard monomials of the Artinian quotient 𝑅𝐼[𝐧]𝑆 is given by the number of irreducible (or indecomposable) permutations of [𝑛+1], while the number of standard monomials of the Artinian quotient 𝑅𝐼[𝐧]𝑇 is given by the number of permutations of [𝑛+1] having no substring {𝑙,𝑙+1}.