Hilbert Polynomial of a Certain Ladder-Determinantal Ideal

A ladder-shaped array is a subset of a rectangular array which looks like a Ferrers diagram corresponding to a partition of a positive integer. The ideals generated by the p-by-p minors of a ladder-type array of indeterminates in the corresponding polynomial ring have been shown to be hilbertian (i.e., their Hilbert functions coincide with Hilbert polynomials for all nonnegative integers) by Abhyankar and Kulkarni 3, p 53–76]. We exhibit here an explicit expression for the Hilbert polynomial of the ideal generated by the two-by-two minors of a ladder-type array of indeterminates in the corresponding polynomial ring. Counting the number of paths in the corresponding rectangular array having a fixed number of turning points above the path corresponding to the ladder is an essential ingredient of the combinatorial construction of the Hilbert polynomial. This gives a constructive proof of the hilbertianness of the ideal generated by the two-by-two minors of a ladder-type array of indeterminates.