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

:(''a''₁, ..., ''a''_''d'')

of nonnegative integers satisfying a system of inequalities

:M\left begina_1 \\ \vdots \\ a_d\end\right\geq \left begin0 \\ \vdots \\ 0\end\right/math>

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

:$F(x_1,\dots,x_d)=\sum_ x_1^\cdots x_d^.$

The generating function ''F''_int(''x''₁, ..., ''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

:$F(1/x_1,\dots,1/x_d)=(-1)^d F_(x_1,\dots,x_d).$

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".

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

See also

* Ehrhart polynomial

References

*
* {{cite arXiv , first1=M. , last1=Beck , first2=M. , last2=Develin , eprint=math.CO/0409562 , title=On Stanley's reciprocity theorem for rational cones , year=2004

Algebraic combinatorics
Theorems in combinatorics