permutohedron

A permutohedron is a geometric object that represents all possible permutations of a set of distinct items. Specifically, it is a convex polytope whose vertices correspond to the permutations of a sequence of numbers, typically 1 to n. Each vertex can be thought of as a point in n-dimensional space, where the coordinates of the point are the permuted numbers. The edges of the permutohedron connect vertices that correspond to permutations that differ by swapping two adjacent elements.

The permutohedron is an n-1 dimensional polytope. For example, the permutohedron for n=3, representing permutations of {1, 2, 3}, is a 2-dimensional hexagon. Its vertices correspond to the permutations (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). The edges connect permutations that can be obtained from each other by a single adjacent transposition.

The permutohedron is closely related to the associahedron and the Birkhoff polytope. It has applications in combinatorics, algebraic geometry, and representation theory. The number of vertices of an n-permutohedron is n!, the number of permutations of n items. The structure of the permutohedron is related to the symmetric group.