In combinatorial mathematics, Stanley's reciprocity theorem, named after MIT mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the generating function of any rational cone (defined below) and the generating function of the cone's interior.
A rational cone is the set of all d-tuples of nonnegative integers satisfying a system of inequalities where M is a matrix of integers.
A d-tuple satisfying the corresponding strict inequalities, i.e., with ">" rather than "≥", is in the interior of the cone.
Stanley's reciprocity theorem states that for a rational cone as above, we have[1] Matthias Beck and Mike Develin have shown how to prove this by using the calculus of residues.
[2] Stanley's reciprocity theorem generalizes Ehrhart-Macdonald reciprocity for Ehrhart polynomials of rational convex polytopes.