• Chanchal Kumar

Articles written in Proceedings – Mathematical Sciences

• 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.

• 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.

• 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.

• 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 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 $\{l, l + 1\}$.

• Integer sequences and monomial ideals

Let $\mathfrak{S}_n$ be the set of all permutations of $[n]=\{1,\ldots , n\}$ and let $W$ be the subset consisting of permutations $\sigma \in \mathfrak{S}_n$ avoiding 132 and 312-patterns. The monomial ideal $I_W = \left\langle \mathbf{x}^{\sigma} = \prod_{i=1}^n x_i^{\sigma(i)} : \sigma \in W \right\rangle$ in the polynomial ring $R = k[x_1,\ldots,x_n]$ over a field $k$ is called a hypercubic ideal in Kumar and Kumar (Proc. Indian Acad. Sci. (Math Sci.) 126(4) (2016) 479--500). The Alexander dual $I_W^{[\mathbf{n}]}$ of $I_W$ with respect to $\mathbf{n}=(n,\ldots,n)$ has the minimal cellular resolution supported on the first barycentric subdivision $\mathbf{Bd}(\Delta_{n-1})$ of an $n−1$-simplex $\Delta_{n-1}$.We show that the number of standard monomials of the Artinian quotient $\frac{R}{I_W^{[\mathbf{n}]}}$ equals the number of rooted-labelled unimodal forests on the vertex set $[n]$. In other words, $$\dim_k\left(\frac{R}{I_W^{[\mathbf{n}]}}\right) = \sum_{r=1}^n r!~s(n,r) = {\rm Per}\left([m_{ij}]_{n \times n} \right),$$where $s(n, r )$ is the (signless) Stirling number of the first kind and ${\rm Per}([m_{ij}]_{n \times n})$ is the permanent of the matrix $[m_{ij}]$ with $m_{ii}=i$ and $m_{ij}=1$ for $i \ne j$. For various subsets $S$ of $\mathfrak{S}_n$ consisting of permutations avoiding patterns, the corresponding integer sequences $\left\lbrace \dim_k\left(\frac{R}{I_S^{[\mathbf{n}]}}\right) \right\rbrace_{n=1}^{\infty}$ are identified.

• # Proceedings – Mathematical Sciences

Volume 132, 2022
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019