  Generalizing a conjecture by De Loera et al., we conjecture that integral
generalized permutohedra all have positive Ehrhart coefficients. Berline and
Vergne construct a valuation that assigns values to faces of polytopes, which
provides a way to write Ehrhart coefficients of a polytope as positive sums of
these values. Based on available results, we pose a stronger conjecture:
Berline-Vergne's valuation is always positive on permutohedra, which implies
our first conjecture.
  This article proves that our strong conjecture on Berline-Vergne's valuation
is true for dimension up to 6, and is true if we restrict to faces of
codimension up to 3. In addition to investigating the positivity conjectures,
we study the Berline-Vergne's valuation, and show that it is the unique
construction for McMullen's formula used to describe number of lattice points
in permutohedra under certain symmetry constraints. We also give an equivalent
statement to the strong conjecture in terms of mixed valuations.
