Artin's reciprocity theorem

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" and the generating function of the cone's interior.

A "rational cone" is the set of all d-tuples

(a1, ..., ad)

of nonnegative integers satisfying a system of inequalities

Mleft[begin{matrix}a_1 vdots a_dend{matrix}right] geq left[begin{matrix}0 vdots 0end{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 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

F(1/x_1,dots,1/x_d)=(-1)^d F_{rm int}(x_1,dots,x_d).

Matthias Beck, Mike Develin, and Sinai Robins 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.

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