Stanley's reciprocity theorem

A rational cone is the set of all d-tuples

:(a₁, ..., a_d)

of nonnegative integers satisfying a system of inequalities

:$M\backslash left[\backslash begin\{matrix\}a\_1\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; a\_d\backslash end\{matrix\}\backslash right]\; \backslash geq\; \backslash left[\backslash begin\{matrix\}0\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; 0\backslash end\{matrix\}\backslash right]$

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

:$F(x\_1,\backslash dots,x\_d)=\backslash sum\_\{(a\_1,\backslash dots,a\_d)\backslash in\; \{\backslash rm\; cone\}\}\; x\_1^\{a\_1\}\backslash cdots\; x\_d^\{a\_d\}.$

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.

Stanley's reciprocity theorem states that for a rational cone as above, we have{{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}}

:$F(1/x\_1,\backslash dots,1/x\_d)=(-1)^d\; F\_\{\backslash rm\; int\}(x\_1,\backslash dots,x\_d).$

Matthias Beck and Mike Develin have shown how to prove this by using the calculus of residues.{{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}}

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

• Ehrhart polynomial

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Category:Algebraic combinatorics

Category:Theorems in combinatorics