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 σv+K(x)≔∑m∈(v+K)∩Zdxm, where KK is a rational cone and v∈Rd, with σ−v+K∘(1/x). The second theorem asserts that the generating function 1+∑n≥1LP(n)tn1+∑n≥1LP(n)tn of the Ehrhart quasi-polynomial LP(n)≔#(nP∩Zd)LP(n)≔#(nP∩Zd) of a rational polytope PP can be written as a rational function νP(t)(1−t)dimP+1 with nonnegative   numerator νPνP. The third theorem asserts that if P⊆QP⊆Q, then νP≤νQνP≤νQ. Our proofs are based on elementary counting afforded by irrational decompositions of rational polyhedra.