Stanley's reciprocity theorem

A rational cone is the set of all d-tuples

(a₁, ..., a_d)

of nonnegative integers satisfying a system of inequalities

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

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

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

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.

Stanley's reciprocity theorem states that for a rational cone as above, we have

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

Matthias Beck and Mike Develin have shown how to prove this by using the calculus of residues. Develin has said that this amounts to proving the result "without doing any work".

Stanley's reciprocity theorem generalizes Ehrhart-Macdonald reciprocity for Ehrhart polynomials of rational convex polytopes.

• Ehrhart polynomial

• Stanley, Richard P. (1974). "Combinatorial reciprocity theorems" (PDF). Advances in Mathematics. 14 (2): 194–253. doi:10.1016/0001-8708(74)90030-9.
• Beck, M.; Develin, M. (2004). "On Stanley's reciprocity theorem for rational cones". arXiv:math.CO/0409562.