  We study the problem of counting lattice points of a polytope that are
weighted by an Ehrhart quasi-polynomial of a family of parametric polytopes. As
applications one can compute integrals and maximum values of such
quasi-polynomials, as well as obtain new identities in representation theory.
These topics have been of great interest to Mich\`ele Vergne since the late
1980's. Our new contribution is a result that transforms weighted sums into
unweighted sums, even when the weights are very general quasipolynomials. In
some cases it leads to faster integration over a polytope. We can create new
algebraic identities and conjectures in algebraic combinatorics and number
theory.
