Irrational proofs for three theorems of Stanley

Details Irrational proofs for three theorems of Stanley Irrational proofs for three theorems of Stanley

2005-01-21

arxiv.org/abs/math/0501359v4

European Journal of Combinatorics 28, no. 1 (2007), 403-409

Mathematics

Combinatorics

7 pages. Minor revisions

Scientific paper

We give new proofs of three theorems of Stanley on generating functions for the integer points in rational cones. The first, Stanley's reciprocity theorem, relates the rational generating functions for the integer points in a cone K and for those in its interior. The second, Stanley's Positivity theorem asserts that the generating function of the Ehrhart quasipolynomial of a rational polytope P can be written as a rational function with nonnegative numerator. The third, Stanley's Monotonicity Theorem, asserts that if a polytope Q contains P, then each coefficient in the numerator for Q is at least as large as the corresponding coefficient in the numerator for P. Our proofs are based on elementary (primary school) counting afforded by irrational decompositions of rational polyhedra.

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