Stanley's reciprocity theorem

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.

Definitions
A rational cone is the set of all d-tuples


 * (a1, ..., ad)

of nonnegative integers satisfying a system of inequalities


 * $$M\left[\begin{matrix}a_1 \\ \vdots  \\ a_d\end{matrix}\right] \geq \left[\begin{matrix}0 \\  \vdots \\  0\end{matrix}\right]$$

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.

The generating function of such a cone is


 * $$F(x_1,\dots,x_d)=\sum_{(a_1,\dots,a_d)\in {\rm cone}} x_1^{a_1}\cdots x_d^{a_d}.$$

The generating function Fint(x1, ..., xd) of the interior of the cone is defined in the same way, but one sums over d-tuples in the interior rather than in the whole cone.

It can be shown that these are rational functions.

Formulation
Stanley's reciprocity theorem states that for a rational cone as above, we have


 * $$F(1/x_1,\dots,1/x_d)=(-1)^d F_{\rm int}(x_1,\dots,x_d).$$

Matthias Beck and Mike Develin have shown how to prove this by using the calculus of residues.

Stanley's reciprocity theorem generalizes Ehrhart-Macdonald reciprocity for Ehrhart polynomials of rational convex polytopes.