Combinatorial meaning of the coefficients of a Hilbert polynomial
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.
scientific article; zbMATH DE number 96604
scientific article