The Reciprocity Theorem

Statement

Stanley's reciprocity theorem states that for a $d$-dimensional rational cone $C$, we have the following identity of rational functions:

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

Stanley's reciprocity theorem generalizes Ehrhart-Macdonald reciprocity for Ehrhart polynomials of rational convex polytopes. Both of these results are examples of combinatorial reciprocity theorems, a term that was, in fact, also coined by Stanley.