Closed-form dimension of weight spaces R_{m,n,ω,τ}

Derive a general closed-form expression for the complex dimension of the weight space R_{m,n,ω,τ}, consisting of degree-d monomials z^a whose exponent matrices a are contingency tables with row margins ω and column margins τ.

Background

Weight spaces R_{m,n,ω,τ} are spanned by monomials with prescribed row and column sums (contingency tables). Counting their dimension reduces to enumerating such tables. The authors note that, in general, computing this dimension is #P-complete (when ω and τ are encoded in binary), and no closed formula is known.