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Stanley's reciprocity theorem

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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

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A rational cone is the set of all d-tuples

(a1, ..., ad)

of nonnegative integers satisfying a system of inequalities

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

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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

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References

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