Suppose I have a (finitely-presented, say) graded module M over k[x,y], and I happen to know the rank $R_{(a,b),(c,d)}$ of each map $x^{c−a}y^{d−b}:M_{a,b}→M_{c,d}$ for each pair of integers with a≤c and b≤d, as well as the Hilbert function $dim_k(M_{a,b})$.

Recall that the betti numbers of M are defined to be the grades and multiplicities of the generators of a minimal free resolution.

Is there an explicit formula which expresses these betti numbers directly in terms of the ranks R_{(a,b),(c,d)} and the hilbert function?

As motivation, there is a well-known expression for the alternating sums of the betti numbers in terms of the hilbert function, e.g. Corollay 1.10 in the Geometry of Syzygies by Eisenbud.