On Hilbert polynomial of certain determinantal ideals (Q1173915)

Let \(X=(X_{ij})\) be an \(m\times n\) matrix of indeterminates over a field \(K\). In the book: ``Enumerative combinatorics of Young tableaux'' (1988; Zbl 0643.05001), \textit{S. S. Abhvankar} considers ideals \(I[p,a]\) in the polynomial ring \(K[X_{ij}| i=1,\ldots,m, j=1,\ldots,n]\) whose systems of generators consist of some combinatorially described minors of \(X\). [For an alternative description of the same ideals see the book by \textit{W. Bruns} and the reviewer ``Determinantal rings'' (1988; Zbl 0673.13006).] He shows that the Hilbert polynomial of these ideals is completely determined by certain integer valued functions. The author proves ``some important properties of these integer valued functions''. He especially gives an affirmative answer to two questions asked in Abhyankar's book.