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, ..., 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,\dots,x_d)=\sum_{(a_1,\dots,a_d)\in {\rm cone}} x_1^{a_1}\cdots x_d^{a_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

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. Develin has said that this amounts to proving the result "without doing any work".

References

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

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