We give new proofs of two theorems of Stanley on generating functions
for the integer points in rational cones.
One is Stanley's reciprocity theorem, relating the generating function at x and
1/x, and the second asserts that the generating
function of the Ehrart quasipolynomial is a rational function with a nonnegataive
numerator.
The proofs are based on elementary (primary school) counting afforded by irrational
decompositions of rational polyhedra.