On a Special Class of Hyper-Permutahedra

Abstract

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 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.