Permutohedron

A Permutaeder is in mathematics, a convex polytope in -dimensional space, the corners of which arise through the permutations of the coordinates of the vector.

Definition

Permutaeder of the order is a convex polytope which is defined as follows: Each permutation of the symmetric group is written as a vector in Tupelschreibweise interpreted. The convex hull of these vectors results in:

The corners of the Permutaeders are precisely the permutations in Tupelschreibweise. Two permutations are exactly connected by an edge of the Permutaeders if they can be converted by a transposition of adjacent elements together.

Properties

The Permutaeder can also be described by the intersection of half-spaces:

The Permutaeder lies in the -dimensional hyperplane

The hyperplane consists precisely of the points whose coordinates sum. She has a tessellation by an infinite number of parallel -shifted copies of the Permutaeders. The symmetry group of this tessellation is given by the following equations -dimensional lattice: