On a Special Class of Hyper-Permutahedra

Geir Agnarsson

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.

 


Keywords


Polytope, Permutahedron, Minkowski sum, Flag polynomial, Exponential flag function.

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