Hilbert polynomials in combinatorics
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.
standard graded commutative \(k\)-algebra
scientific article; zbMATH DE number 1139609
scientific article