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 $a\leq c$ and $b \leq d$, 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.