Editing Stanley's reciprocity theorem

{{short description|Gives a functional equation satisfied by the generating function of any rational cone}}
In [[combinatorics|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''-[[tuple]]s

:(''a''<sub>1</sub>, ..., ''a''<sub>''d''</sub>) 

of [[nonnegative integer]]s satisfying a [[system of inequalities]]

:<math>M\left[\begin{matrix}a_1 \\  \vdots  \\ a_d\end{matrix}\right] \geq \left[\begin{matrix}0 \\  \vdots \\  0\end{matrix}\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

:<math>F(x_1,\dots,x_d)=\sum_{(a_1,\dots,a_d)\in {\rm cone}} x_1^{a_1}\cdots x_d^{a_d}.</math>

The generating function ''F''<sub>int</sub>(''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>) 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 function]]s.

== Formulation ==
Stanley's reciprocity theorem states that for a rational cone as above, we have<ref>{{cite journal |first=Richard P. |last=Stanley |title=Combinatorial reciprocity theorems |journal=[[Advances in Mathematics]] |volume=14 |issue=2 |pages=194–253 |year=1974 |doi=10.1016/0001-8708(74)90030-9 |doi-access=free |url=http://math.mit.edu/~rstan/pubs/pubfiles/22.pdf}}</ref>

:<math>F(1/x_1,\dots,1/x_d)=(-1)^d F_{\rm int}(x_1,\dots,x_d).</math>

[[Matthias Beck]] and [[Mike Develin]] have shown how to prove this by using the [[methods of contour integration|calculus of residues]].<ref>{{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}}</ref>

Stanley's reciprocity theorem generalizes Ehrhart-Macdonald reciprocity for [[Ehrhart polynomials]] of rational [[Convex polytope|convex polytopes]].

==See also==
* [[Ehrhart polynomial]]

==References==
{{reflist}}


[[Category:Algebraic combinatorics]]
[[Category:Theorems in combinatorics]]