Irrational proofs for three theorems of Stanley

We give new proofs of three theorems of Stanley on generating functions for the integer points in rational cones. The first relates the rational generating function σ v + K ( x ) ≔ ∑ m ∈ ( v + K ) ∩ Z d x m , where K is a rational cone and v ∈ R d , with σ − v + K ∘ ( 1 / x ) . The second theorem asserts that the generating function 1 + ∑ n ≥ 1 L P ( n ) t n of the Ehrhart quasi-polynomial L P ( n ) ≔ # ( n P ∩ Z d ) of a rational polytope P can be written as a rational function ν P ( t ) ( 1 − t ) dim P + 1 with nonnegative numerator ν P . The third theorem asserts that if P ⊆ Q , then ν P ≤ ν Q . Our proofs are based on elementary counting afforded by irrational decompositions of rational polyhedra.