Permutohedra

A permutohedron is a geometric object that represents all possible permutations of a set of distinct numbers. Specifically, it is the convex hull of the points obtained by permuting the coordinates of a vector. For example, if we consider the vector (1, 2, 3), the permutations of its coordinates are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). The permutohedron is the three-dimensional shape formed by connecting these points.

In general, for a vector with n distinct elements, the permutohedron is an (n-1)-dimensional polytope. The vertices of an n-dimensional permutohedron correspond to the n! possible permutations of the coordinates of the base vector. The edges of a permutohedron connect vertices that differ by swapping two adjacent coordinates.

Permutohedra have connections to various areas of mathematics, including combinatorics, representation theory, and algebraic geometry. They can be viewed as a special case of associahedra, which represent the possible ways to parenthesize a product of elements. The study of permutohedra is important for understanding symmetries and combinatorial structures.