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Stanley's reciprocity theorem

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Gives a functional equation satisfied by the generating function of any rational cone

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

M [ a 1 ⋮ a d ] ≥ [ 0 ⋮ 0 ] {\displaystyle M\left\geq \left}

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 , … , x d ) = ∑ ( a 1 , … , a d ) ∈ c o n e x 1 a 1 ⋯ x d a d . {\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 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

F ( 1 / x 1 , … , 1 / x d ) = ( − 1 ) d F i n t ( x 1 , … , x d ) . {\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.

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

See also

References

  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.
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