On a special class of general permutahedra

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 ${\mathrm{\Delta }}_{k-1}$ 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 ${\mathrm{\Delta }}_{n-1}$ and the permutahedron ${\mathrm{\Pi }}_{n-1}$, and form a Minkowski basis for more general permutahedra.