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.
A rational cone is the set of all d-tuples
- (a1, ..., ad)
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.
Stanley's reciprocity theorem states that
- R.P. Stanley, "Combinatorial reciprocity theorems", Advances in Mathematics, volume 14 (1974), pages 194–253.
- M. Beck, M. Develin, On Stanley's reciprocity theorem for rational cones, 2004