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| == We're working on Call of Duty Timberland Schuhe ==
| | 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. |
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| | A '''rational cone''' is the set of all ''d''-[[tuple]]s |
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| | :(''a''<sub>1</sub>, ..., ''a''<sub>''d''</sub>) |
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| | of [[nonnegative integer]]s satisfying a [[system of inequalities]] |
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| == we Louboutin Berlin ==
| | :<math>M\left[\begin{matrix}a_1 \\ \vdots \\ a_d\end{matrix}\right] \geq \left[\begin{matrix}0 \\ \vdots \\ 0\end{matrix}\right]</math> |
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| | 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. |
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| == 23 in the journal Cancer Abercrombie Deutschland ==
| | The generating function of such a cone is |
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| | :<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> |
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| | 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. |
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| | It can be shown that these are [[rational function]]s. Stanley's reciprocity theorem states that |
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| | :<math>F(1/x_1,\dots,1/x_d)=(-1)^d F_{\rm int}(x_1,\dots,x_d).</math> |
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| | [[Matthias Beck]], [[Mike Develin]], and [[Sinai Robins]] have shown how to prove this by using the [[methods of contour integration|calculus of residues]]. Develin has said that this amounts to proving the result "without doing any work".{{fact|date=March 2010}} |
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| | ==References== |
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| </ul>
| | * R.P. Stanley, "Combinatorial reciprocity theorems", ''[[Advances in Mathematics]]'', volume 14 (1974), pages 194–253. |
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| | ==External links== |
| | * [http://arxiv.org/abs/math.CO/0409562 On Stanley's reciprocity theorem for rational cones, by Beck, Develin, and Robins -- arXiv abstract] |
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| | [[Category:Algebraic combinatorics]] |
| | [[Category:Theorems in combinatorics]] |
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.
A rational cone is the set of all d-tuples
- (a1, ..., ad)
of nonnegative integers satisfying a system of inequalities
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
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
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".Template:Fact
References
External links