  We consider the Ehrhart polynomial of hypersimplices. It is proved that these
polynomials have positive coefficients and we give a combinatorial formula for
each of them. This settles a problem posed by Stanley and also proves that
uniform matroids are Ehrhart positive, an important and yet unsolved particular
case of a conjecture posed by De Loera et al. To this end, we introduce a new
family of numbers that we call weighted Lah numbers and study some of their
properties.
