• Corpus ID: 15003121

# On Stanley's reciprocity theorem for rational cones

@article{Beck2004OnSR,
title={On Stanley's reciprocity theorem for rational cones},
author={Matthias Beck and Mike Develin},
journal={arXiv: Combinatorics},
year={2004}
}
• Published 28 September 2004
• Mathematics
• arXiv: Combinatorics
We give a short, self-contained proof of Stanley's reciprocity theorem for a rational cone K \subset R^d. Namely, let sigma_K (x) = sum_{m \in K \cap Z^d} x^m. Then sigma_K (x) and sigma_int(K) (x) are rational functions which satisfy the identity sigma_K (1/x) = (-1)^d sigma_int(K) (x). A corollary of Stanley's theorem is the Ehrhart-Macdonald reciprocity theorem for the lattice-point enumerator of rational polytopes. A distinguishing feature of our proof is that it uses neither the shelling…
1 Citations

## References

SHOWING 1-10 OF 18 REFERENCES
Lattice invariant valuations on rational polytopes
AbstractLetΛ be a lattice ind-dimensional euclidean space $$\mathbb{E}^d$$ , and $$\bar \Lambda$$ the rational vector space it generates. Ifϕ is a valuation invariant underΛ, andP is a polytope
Combinatorics and commutative algebra
This text offers an overview of two of the main topics in the connections between commutative algebra and combinatorics. The first concerns the solutions of linear equations in non-negative integers.
Combinatorial reciprocity theorems
A combinatorial reciprocity theorem is a result which establishes a kind of duality between two related enumeration problems. This rather vague concept will become clearer as more and more examples
De partitione numerorum in partes tam numero quam specie dates, Leonhardi Euleri Opera Omnia
• De partitione numerorum in partes tam numero quam specie dates, Leonhardi Euleri Opera Omnia
• 1917
E-mail address: beck@math.sfsu.edu URL: http://math
• E-mail address: beck@math.sfsu.edu URL: http://math
Finally, there exists an extension of Theorem 4 corresponding to Theorem 3: one includes some of the facets of the polytope on one side, and the complementary set of facets on the other side
• Finally, there exists an extension of Theorem 4 corresponding to Theorem 3: one includes some of the facets of the polytope on one side, and the complementary set of facets on the other side
Sur les polyèdres rationnels homothétiques à n dimensions
• Acad . Sci . Paris
Combinatorial reciprocity theorems, Advances in Math
• Combinatorial reciprocity theorems, Advances in Math
• 1974