permutahedron

A permutahedron is a polytope, or multidimensional geometric shape, that generalizes the concept of permutations to higher dimensions. Specifically, it is the convex hull of all permutations of a given set of points in Euclidean space. The name derives from the Latin *permutatio*, meaning "change" or "rearrangement," reflecting the fundamental role of permutations in its construction.

In two dimensions, a permutahedron corresponds to a convex polygon whose vertices are the permutations of a set of points on a line. For example, with three points, the permutahedron is a triangle formed by the three possible orderings of the points. In three dimensions, a permutahedron is a polyhedron whose vertices represent all possible orderings of a set of points along a line. The simplest non-trivial case involves four points, yielding a polyhedron with 24 vertices (since there are 4! = 24 permutations of four elements). This shape is known as the *associahedron* when considering non-crossing partitions, but in the general case, it is a permutahedron.

Permutahedra are closely related to the *Bruck-Ryser-Chowla theorem* and have applications in combinatorics, particularly in the study of Catalan numbers and non-crossing partitions. They also appear in the context of hyperplane arrangements and the study of the *permutation group* acting on a set of points. The facets of a permutahedron correspond to inequalities derived from the relative ordering of points, making them useful in optimization and computational geometry.

The number of vertices of an *n*-dimensional permutahedron is given by the factorial of *n*, as each vertex represents a unique permutation of *n* elements. The structure of permutahedra has been extensively studied in algebraic topology, where they serve as examples of simplicial complexes and toric varieties. Their geometric properties, such as edge lengths and dihedral angles, depend on the specific arrangement of the original points.