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, ..., ad)
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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c111f7af4e3f48d5efa7e6b2530c71e919f08f9c)
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
The generating function Fint(x1, ..., xd) 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[1]
Matthias Beck and Mike Develin have shown how to prove this by using the calculus of residues.[2]
Stanley's reciprocity theorem generalizes Ehrhart-Macdonald reciprocity for Ehrhart polynomials of rational convex polytopes.
See also
References
- ^ 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.