  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.
