Other properties Permutohedron

cayley graph of s4, generated 3 adjacent transpositions of 4 elements

only self-inverse permutations @ same positions in permutohedron; others replaced inverses.

the permutohedron vertex-transitive: symmetric group sn acts on permutohedron permutation of coordinates.

the permutohedron zonotope; translated copy of permutohedron can generated minkowski sum of n(n − 1)/2 line segments connect pairs of standard basis vectors .

the vertices , edges of permutohedron isomorphic undirected graph 1 of cayley graphs of symmetric group: cayley graph generated adjacent transpositions in symmetric group (transpositions swap consecutive elements). cayley graph of s4, shown on right, generated transpositions (1,2), (2,3), , (3,4).

the cayley graph labeling can constructed labeling each vertex inverse of permutation given coordinates.

this cayley graph hamiltonian; hamiltonian cycle may found steinhaus–johnson–trotter algorithm.