Stanley's reciprocity theorem

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

$M\left[{\begin{matrix}a_{1}\\\vdots \\a_{d}\end{matrix}}\right]\geq \left[{\begin{matrix}0\\\vdots \\0\end{matrix}}\right]$

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

$F(x_{1},\dots ,x_{d})=\sum _{(a_{1},\dots ,a_{d})\in {\rm {cone}}}x_{1}^{a_{1}}\cdots x_{d}^{a_{d}}.$

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

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

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