Counting of paths and coefficients of the Hilbert polynomial of a determinantal ideal Original Research Article

Let X= [Xij]1⩽i⩽m1⩽i⩽n be an m × n of matrix of indeterminates over a field K. Abhyankar defines the index of a monomial in Xij to be the largest k such that the principal diagonal of some k × k minor of X divides the given monomial. Abhyankar has given a formula for counting the set of monomials in Xij of degree v of index at most p, satisfying a certain set of index conditions. This formula gives the Hilbert polynomial of a certain generalized determinantal ideal which can be viewed as a polynomial in v with rational coefficients. We develop a combinatorial map from this set of monomials to the set of p-tuples of nonintersecting paths in the m × n rectangular lattice of points. A path from (a, n) to (m, b) in a rectangular m × n array is obtained by moving either left or down at each point. The point where the path turns from down to left is called its node. Using the combinatorial map, we get formulae counting sets of p-tuples of nonintersecting paths having a fixed number of nodes in a rectangular lattice. This helps us to interpret ‘coefficients’ of Hilbert polynomials of generalized determinantal ideals combinatorially. This enables us to answer questions raised by Abhyankar in the monograph ‘Enumerative Combinatorics of Young Tableaux’.