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An integer sequence and standard monomials

Ajay Kumar

DAV University, Jalandhar, Punjab - 144 012, India

E-mail Address: ajaychhabra.msc@gmail.com

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and

Chanchal Kumar

IISER Mohali, Knowledge City Sector 81, SAS Nagar, Punjab -140 306, India

E-mail Address: chanchal@iisermohali.ac.in

Corresponding author.

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https://doi.org/10.1142/S0219498818500378Cited by:1 (Source: Crossref)

Abstract

For an (oriented) graph $G$ on the vertex set ${0, 1, \dots, n}$ (rooted at $0$ ), Postnikov and Shapiro (Trans. Amer. Math. Soc.356 (2004) 3109–3142) associated a monomial ideal $ℳ_{G}$ in the polynomial ring $R = k [x_{1}, \dots, x_{n}]$ over a field $k$ such that the number of standard monomials of $\frac{R}{ℳ_{G}}$ equals the number of (oriented) spanning trees of $G$ and hence, ${dim}_{k} (\frac{R}{ℳ_{G}}) = det (L_{G})$ , where $L_{G}$ is the truncated Laplace matrix of $G$ . The standard monomials of $\frac{R}{ℳ_{G}}$ correspond bijectively to the $G$ -parking functions. In this paper, we study a monomial ideal $J_{n}$ in $R$ having rich combinatorial properties. We show that the minimal free resolution of the monomial ideal $J_{n}$ is the cellular resolution supported on a subcomplex of the first barycentric subdivision $B d (Δ_{n - 1})$ of an $n - 1$ simplex $Δ_{n - 1}$ . The integer sequence ${{dim}_{k} (\frac{R}{J_{n}})}_{n \geq 1}$ has many interesting properties. In particular, we obtain a formula, ${dim}_{k} (\frac{R}{J_{n}}) = det ({[m_{i j}]}_{n \times n})$ , with $m_{i j} = 1$ for $i > j$ , $m_{i i} = i$ and $m_{i j} = i - j$ for $i < j$ , similar to ${dim}_{k} (\frac{R}{ℳ_{G}}) = det (L_{G})$ .

Communicated by T. Ha

Keywords:

AMSC: 05E40, 13D02

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