Hilbert polynomials in combinatorics
type: Item
instance of: scholarly article
title: Hilbert polynomials in combinatorics
zbMATH Open document ID: 0901.05093
DOI: 10.1023/A:1008656320759
author: Francesco Brenti
published in: Journal of Algebraic Combinatorics
publication date: 1998-11-17T00:00:00Z
review text: The paper investigates which of the polynomials naturally arising in combinatorics are Hilbert polynomials of standard graded commutative \(k\)-algebras (\(k\) being some field). Using Macaulay's combinatorial characterization of Hilbert functions and polynomials as well as other techniques, it is shown that the following polynomials are Hilbert polynomials (up to slight modifications in some cases): The \(\sigma\)- and \(\tau\)-polynomials of a graph, the zeta polynomial of a partially ordered set, the \(R\)-polynomial of two generic elements in a Coxeter system, the Kazhdan-Lusztig polynomials and the descent generating function of a finite Coxeter system, various generalizations of the Eulerian polynomials related to Stirling multi-permutations, Stirling polynomials and several polynomials obtained by specializing certain symmetric functions.
reviewed by: Markus Fulmek
Mathematics Subject Classification ID: 05E99
zbMATH DE Number: 1139609
zbMATH Keywords: standard graded commutative \(k\)-algebra
zbMATH Keywords: Hilbert function
zbMATH Keywords: Hilbert polynomial
zbMATH Keywords: chromatic polynomial
zbMATH Keywords: Coxeter system
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1139609