# Permutohedron

The *n*th-order **permutohedron** (sometimes **permutahedron**) is a convex (*n* - 1)-polytope whose vertex coordinates are all possible permutations of (1, 2, ..., *n*). Although these vectors are *n*-dimensional, 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. Despite the suffix, only the 3-dimensional permutohedron is a polyhedron; a more accurate name would have been "permutotope."

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). In fact the *n*th-order permutohedron is (a variant of) the (*n*-1)-dimensional omnitruncated simplex and has *n*! vertices, (*n* - 1) *n*! / 2 edges, and facets of rank *d* where the curly braces denote the Stirling numbers of the second kind.

All permutohedra can tile space, and all are uniform.

Permutohedra are closely related to the associahedra.