Stanley's reciprocity theorem
Stanley's reciprocity theorem, named after the mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the integer-point generating function of a rational cone and the generating function of the cone's interior.
Definitions
A rational cone is a subset of consisting of all points satisfying a finite set of homogeneous linear inequalities with integer coefficients or, alternatively, the nonnegative span of a finite set of integer vectors. That is, a rational cone has the two alternative descriptionsfor some integer matrix, and
for some integer matrix .
The integer-point generating function of such a cone is
The generating function of the interior of the cone is defined analogously.
It can be shown that these generating functions evaluate to rational functions.
The Reciprocity Theorem
Stanley's reciprocity theorem states that for a -dimensional rational cone, we have the following identity of rational functions: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.