Stanley's reciprocity theorem

Template:Short description 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 ${\mathbf{𝐑}}^d$ 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 C has the two alternative descriptions

$C=\{x\in {\mathbf{𝐑}}^d:Ax\le 0\}$

for some $m\times d$ integer matrix A (i.e., C is defined by the m halfspaces given by the rows of A), and

$C=\{By:y\ge 0\}$

for some $d\times n$ integer matrix B (i.e., C is defined as the nonnegative span of the n columns of B).

The integer-point generating function (also called integer-point transform) of such a cone C is

$F_C(x_1,\dots ,x_d)=\sum _{(a_1,\dots ,a_d)\in C\cap {\mathbf{𝐙}}^d}{x}_1^{a_1}\cdots x_d^{a_d}.$

The generating function $F_{\mathrm{i}\mathrm{n}\mathrm{t}(C)}(x_1,\dots ,x_d)$ 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 $d$-dimensional rational cone $C$, we have the following identity of rational functions:

$F_C(1/x_1,\dots ,1/x_d)=(-1{)}^d{F}_{\mathrm{i}\mathrm{n}\mathrm{t}(C)}(x_1,\dots ,x_d).$

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.

See also

• Ehrhart polynomial