Stanley's reciprocity theorem

A rational cone is a subset of ${\displaystyle {\bf {R}}^{d}}$ 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

${\displaystyle C=\left\{x\in {\bf {R}}^{d}:Ax\leq 0\right\}}$

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

${\displaystyle C=\left\{By:y\geq 0\right\}}$

for some ${\displaystyle d\times n}$ 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

${\displaystyle F_{C}(x_{1},\dots ,x_{d})=\sum _{(a_{1},\dots ,a_{d})\in C\cap {\bf {Z}}^{d}}x_{1}^{a_{1}}\cdots x_{d}^{a_{d}}.}$

The generating function ${\displaystyle F_{{\rm {int}}(C)}(x_{1},\dots ,x_{d})}$ of the interior of the cone is defined analogously. It can be shown that these generating functions evaluate to rational functions.

Stanley's reciprocity theorem states that for a ${\displaystyle d}$-dimensional rational cone ${\displaystyle C}$, we have the following identity of rational functions:^[1]

${\displaystyle F_{C}(1/x_{1},\dots ,1/x_{d})=(-1)^{d}F_{{\rm {int}}(C)}(x_{1},\dots ,x_{d}).}$

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,^[2] a term that was, in fact, also coined by Stanley.

• Ehrhart polynomial