Combinatorial meaning of the coefficients of a Hilbert polynomial
type: Item
instance of: scholarly article
title: Combinatorial meaning of the coefficients of a Hilbert polynomial
zbMATH Open document ID: 0761.13006
DOI: 10.1007/BF02836782
author: M. R. Modak
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1993-01-17T00:00:00Z
review text: Let \(X\) be a matrix of independent indeterminates \(X_{ij}\) over a field \(K\), and let \(K[X]\) be the ring of polynomials in these indeterminates. In the book ``Enumerative combinatorics of Young tableaux'' (New York 1988; Zbl 0643.05001), \textit{S. S. Abhyankar} obtained a formula \(F(V)=\sum_{D\geq 0}(-1)^ D\cdot F_ D\cdot{V+C-D\choose V}\) for the Hilbert function of an ideal of \(K[X]\) generated by certain minors of \(X\). He showed \(F(V)\) to be the cardinality of a suitable set of (indexed) monomials. The present author establishes a relation between these monomials and certain sets of lattice paths, called frames. In response to a question of Abhyankar, he obtains combinatorial interpretations of the coefficients \(F_ D\) in terms of frames.
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 05C38
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 96604
zbMATH Keywords: Hilbert function of polynomial ideal
zbMATH Keywords: lattice paths
zbMATH Keywords: frames
reviewed by: C. P. L. Rhodes
MaRDI profile type: Publication
cites work: Q4039722
cites work: On Hilbertian ideals
cites work: Binomial determinants, paths, and hook length formulae
cites work: Q4295418
cites work: On Hilbert polynomial of certain determinantal ideals
Recommended article: Hilbert polynomials in combinatorics
Recommended article: Q4316782
Recommended article: On the coefficients of the Hilbert polynomial
Recommended article: A note on the coefficients of Hilbert polynomial
Recommended article: Combinatorics of F-Polynomials
Recommended article: On the coefficients of Hilbert quasipolynomials
Recommended article: The combinatorics of associated Hermite polynomials
Recommended article: A Combinatorial Interpretation of the Chebyshev Polynomials
Recommended article: A combinatorial interpretation of the Conti-Contucci-Falcolini polynomial
Recommended article: Combinatorial bases of polynomials
description: scientific article; zbMATH DE number 96604