Stanley's reciprocity theorem

Template:Short description 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, ..., a_d)

of nonnegative integers satisfying a system of inequalities

$M\left[\begin{array}{c}{a}_1\\ ⋮\\ a_d\end{array}\right]\ge \left[\begin{array}{c}0\\ ⋮\\ 0\end{array}\right]$

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

$F(x_1,\dots ,x_d)=\sum _{(a_1,\dots ,a_d)\in \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}}{x}_1^{a_1}\cdots x_d^{a_d}.$

The generating function F_int(x₁, ..., x_d) 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

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

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.

See also

• Ehrhart polynomial