permutahedra

A permutohedron is a geometric object studied in combinatorics and discrete geometry. It is a specific type of convex polytope. The permutohedron of order n, denoted P_n, is the convex hull of all points that can be obtained by permuting the coordinates of the vector (1, 2, ..., n). In other words, if you take the set of all possible orderings of the numbers from 1 to n, and consider each ordering as a point in n-dimensional space, the permutohedron is the smallest convex set containing all these points.

The vertices of the permutohedron P_n correspond to the permutations of the set {1, 2, ..., n}. The number of vertices is n!, the factorial of n. The edges of the permutohedron connect vertices that correspond to permutations that differ by a single transposition (swapping two adjacent elements). The permutohedron is a centrally symmetric polytope. It can also be described as the set of points (x_1, x_2, ..., x_n) in R^n such that the sum of the coordinates is fixed (specifically, the sum is n(n+1)/2) and the ordered coordinates satisfy certain inequalities. The permutohedron is closely related to the associahedron and the Birkhoff polytope.