Monomial ideals induced by permutations avoiding patterns
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Let $S$ (or $T$ ) be the set of permutations of $[n] = \{1, . . . , n\}$ avoiding123 and 132 patterns (or avoiding 123, 132 and 213 patterns). The monomial ideals $I_{S} = \langle\rm{x}^\sigma = \prod^{n}_{i=1}x^{\sigma(i)}_{i} : \sigma \in S\rangle$ and $I_{T} = \langle\rm{x}^{\sigma} : \sigma \in T \rangle$ in the polynomial ring$R = k[x_{1}, . . . , x_{n}]$ over a field $k$ have many interesting properties. The Alexander dual $I^{[n]}_{S}$ of $I_{S}$ with respect to $\bf{n} = (n, . . . , n)$ has the minimal cellular resolution supported on the order complex $\Delta(\Sigma_{n})$ of a poset $\Sigma_{n}$. The Alexander dual $I^{[n]}_{T}$ also has the minimalcellular resolution supported on the order complex $\Delta(\tilde{\Sigma}_{n})$ of a poset $\tilde{\Sigma}_{n}$. The number of standard monomials of the Artinian quotient $\frac{R}{I^{[n]}_{S}}$ is given by the number of irreducible(or indecomposable) permutations of $[n + 1]$, while the number of standard monomials of the Artinian quotient $\frac{R}{I^{[n]}_{T}}$is given by the number of permutations of $[n + 1]$ having no substring $\{l, l + 1\}$.
AJAY KUMAR^{1} ^{} CHANCHAL KUMAR^{2} ^{}
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Volume 129 | Issue 5
November 2019
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