  Minkowski sums of simplices in ${\mathbb{R}}^n$ form an interesting class of
polytopes that seem to emerge in various situations. In this paper we discuss
the Minkowski sum of the simplices $\Delta_{k-1}$ in ${\mathbb{R}}^n$ where $k$
and $n$ are fixed, their flags and some of their face lattice structure. In
particular, we derive a closed formula for their {\em exponential generating
flag function}. These polytopes are simple, include both the simplex
$\Delta_{n-1}$ and the permutahedron $\Pi_{n-1}$, and form a Minkowski basis
for more general permutahedra.
