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Monomial ideals induced by permutations avoiding patterns

  • Ajay Kumar
  • Chanchal KumarEmail author
Article
  • 16 Downloads

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

Keywords

Permutations avoiding patterns cellular resolutions standard monomials parking functions 

1991 Mathematics Subject Classification

05E40 13D02 

Notes

Acknowledgements

Thanks are due to the anonymous referee for many valuable suggestions that improved the overall presentation of the paper.

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Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.DAV UniversityJalandharIndia
  2. 2.IISER MohaliMohaliIndia

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