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We show how to compute the Ehrhart polynomial of the free sum of two latticepolytopes containing the origin $P$ and $Q$ in terms of the enumerativecombinatorics of $P$ and $Q$. This generalizes work of Beck, Jayawant,McAllister, and Braun, and follows from the observation that the weighted$h^*$-polynomial is multiplicative with respect to the free sum. We deduce thatgiven a lattice polytope $P$ containing the origin, the problem of computingthe number of lattice points in all rational dilates of $P$ is equivalent tothe problem of computing the number of lattice points in all integer dilates ofall free sums of $P$ with itself.

Counting lattice points in free sums of polytopes