Permutohedron

The n th-order permutohedron or permutahedron is a convex (n - 1)-polytope whose vertex coordinates are all possible permutations of (1, 2, ..., n). The n-th order permutohedron is the uniform omnitruncated (n - 1)-simplex.

Although the permutohedron is defined as the convex hull of vectors in n -dimensional space, they all fall in a (n - 1)-hyperplane as the sum of the coordinates is constant. Two vertices are connected by an edge if they are related by swapping exactly two coordinates whose values differ by one.

The first few permutohedra are the point (order 1), line segment (order 2), regular hexagon (order 3), truncated octahedron (order 4), great prismatodecachoron (order 5), and great cellidodecateron (order 6). The n-th order permutohedron has n! vertices, (n - 1) n! / 2 edges, and $(n-d)!\left\{{n \atop n-d}\right\}$ facets of rank d  where the curly braces denote the Stirling numbers of the second kind.

All permutohedra can tile space.

See also

• Associahedron