On a special class of general permutahedra

TL;DR

This paper studies a special class of polytopes formed by Minkowski sums of simplices, deriving formulas for their face structures and highlighting their role as a basis for more complex permutahedra.

Contribution

It provides a closed-form formula for the exponential generating flag function of these polytopes and explores their face lattice structure.

Findings

Derived a closed formula for the exponential generating flag function.

Identified these polytopes as a Minkowski basis for general permutahedra.

Showed these polytopes include the simplex and permutahedron as special cases.

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 {\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.