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… Expand

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… Expand

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.… Expand

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… Expand

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