CHANCHAL KUMAR
Articles written in Proceedings – Mathematical Sciences
Volume 120 Issue 2 April 2010 pp 163-168
Deficiently Extremal Cohen-Macaulay Algebras
The aim of this paper is to study homological properties of deficiently extremal Cohen–Macaulay algebras. Eagon–Reiner showed that the Stanley–Reisner ring of a simplicial complex has a linear resolution if and only if the Alexander dual of the simplicial complex is Cohen–Macaulay. An extension of a special case of Eagon–Reiner theorem is obtained for deficiently extremal Cohen–Macaulay Stanley–Reisner rings.
Volume 124 Issue 1 February 2014 pp 1-15
Alexander Duals of Multipermutohedron Ideals
An Alexander dual of a multipermutohedron ideal has many combinatorial properties. The standard monomials of an Artinian quotient of such a dual correspond bijectively to some 𝜆-parking functions, and many interesting properties of these Artinian quotients are obtained by Postnikov and Shapiro (Trans. Am. Math. Soc. 356 (2004) 3109–3142). Using the multigraded Hilbert series of an Artinian quotient of an Alexander dual of multipermutohedron ideals, we obtained a simple proof of Steck determinant formula for enumeration of 𝜆-parking functions. A combinatorial formula for all the multigraded Betti numbers of an Alexander dual of multipermutohedron ideals are also obtained.
Volume 126 Issue 4 October 2016 pp 479-500 Research Article
Certain variants of multipermutohedron ideals
Multipermutohedron ideals have rich combinatorial properties. An explicit combinatorial formula for the multigraded Betti numbers of a multipermutohedron ideal and their Alexander duals are known. Also, the dimension of the Artinian quotient of an Alexander dual of a multipermutohedron ideal is the number of generalized parking functions. In this paper, monomial ideals which are certain variants of multipermutohedron ideals are studied. Multigraded Betti numbers of these variant monomial ideals and their Alexander duals are obtained. Further, many interesting combinatorial properties of multipermutohedron ideals are extended to these variant monomial ideals.
Volume 129 Issue 1 February 2019 Article ID 0010 Research Article
Monomial ideals induced by permutations avoiding patterns
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
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Volume 129 | Issue 2
April 2019
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