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} }
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…
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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
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