Stanley's reciprocity theorem explained

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, ..., a_d)

of nonnegative integers satisfying a system of inequalities

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

F(x _{1,...,x} _{d)=\sum} ^{(a _{1,...,a} _{d)\in} {\rm cone} } x_1^\cdots x_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.

Formulation

Stanley's reciprocity theorem states that for a rational cone as above, we have^[1]

F( 1 /x _{1,...,1/x} ^{d} ^{d) = ( -1 )} F _{\rm} (x _{1,...,x} _{d).}

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

• Ehrhart polynomial

Notes and References

1. Richard P. . Stanley . Combinatorial reciprocity theorems . . 14 . 2 . 194–253 . 1974 . 10.1016/0001-8708(74)90030-9 . free .
2. M. . Beck . M. . Develin . math.CO/0409562 . On Stanley's reciprocity theorem for rational cones . 2004.