Permutohedron

The permutohedron of order 4

In mathematics, the permutohedron of order n (also spelled permutahedron)[1] is an (n − 1)-dimensional polytope embedded in an n-dimensional space, the vertices of which are formed by permuting the coordinates of the vector (1, 2, 3, ..., n).

History

According to Ziegler (1995), permutohedra were first studied by Schoute (1911). The name "permutohedron" (or rather its French version, "permutoèdre") was coined by Guilbaud & Rosenstiehl (1963). Regarding this coinage, they write that the word "permutohedron" is barbaric, but easy to remember, and that they submit it to the criticism of their readers.[2]

The alternative spelling permutahedron is sometimes also used. Permutohedra are sometimes also called permutation polytopes, but this terminology is also used for a related polytope, the Birkhoff polytope, defined as the convex hull of permutation matrices. More generally, Bowman (1972) uses the phrase "permutation polytope" for any polytope whose vertices are in 1-1 correspondence with the permutations of some set.

Vertices, edges, and facets

The 13 weak orderings on (or ordered partitions of) the set {1,2,3} in the permutohedron-like Cayley graph of S3
   k = 1    2    3    4    5
n
1      1                               1
2      1    2                          3
3      1    6    6                    13
4      1   14   36   24               75
5      1   30  150  240  120         541

The permutohedron of order n has n! vertices, each of which is adjacent to n − 1 others, so the total number of edges is (n − 1)n!/2. Each edge has length √2, and connects two vertices that differ by swapping two coordinates the values of which differ by one.[3]

The permutohedron has one facet for each nonempty proper subset S of {1, 2, 3, ..., n}, consisting of the vertices in which all coordinates in positions in S are smaller than all coordinates in positions not in S. Thus, the total number of facets is 2n − 2. More generally, the faces of the permutohedron (including the permutohedron itself, but not including the empty set) are in 1-1 correspondence with the strict weak orderings on a set of n items: a face of dimension d corresponds to a strict weak ordering in which there are n − d equivalence classes. Because of this correspondence, the number of faces is given by the ordered Bell numbers.[4]

The number of (n−k)-dimensional faces in a permutohedron of order n is found in the triangle T(n,k) = k! * Stirling2(n,k) (sequence A019538 in the OEIS) - shown on the right, together with its row sums, the ordered Bell numbers.

Other properties

Cayley graph of S4, generated by the 3 adjacent transpositions of 4 elements
Only self-inverse permutations are at the same positions as in the permutohedron; the others are replaced by their inverses.

The permutohedron is vertex-transitive: the symmetric group Sn acts on the permutohedron by permutation of coordinates.

The permutohedron is a zonotope; a translated copy of the permutohedron can be generated as the Minkowski sum of the n(n − 1)/2 line segments that connect the pairs of the standard basis vectors .[5]

The vertices and edges of the permutohedron are isomorphic as an undirected graph to one of the Cayley graphs of the symmetric group: The Cayley graph generated by the adjacent transpositions in the symmetric group (transpositions that swap consecutive elements). The Cayley graph of S4, shown on the right, is generated by the transpositions (1,2), (2,3), and (3,4).
The Cayley graph labeling can be constructed by labeling each vertex by the inverse of the permutation given by its coordinates.[6]

This Cayley graph is Hamiltonian; a Hamiltonian cycle may be found by the Steinhaus–Johnson–Trotter algorithm.

Tessellation of the space

Tesselation of space by permutohedra

The permutohedron of order n lies entirely in the (n − 1)-dimensional hyperplane consisting of all points whose coordinates sum to the number

1 + 2 + … + n = n(n + 1)/2.

Moreover, this hyperplane can be tiled by infinitely many translated copies of the permutohedron. Each of them differs from the basic permutohedron by an element of a certain (n − 1)-dimensional lattice, which consists of the n-tuples of integers that sum to zero and whose residues (modulo n) are all equal:

x1 + x2 + … + xn = 0,     x1 ≡ x2 ≡ … ≡ xn (mod n).

Thus, the permutohedron of order 4 shown above tiles the 3-dimensional space by translation. Here the 3-dimensional space is the affine subspace of the 4-dimensional space R4 with coordinates x, y, z, w that consists of the 4-tuples of real numbers whose sum is 10,

x + y + z + w = 10.

One easily checks that for each of the following four vectors,

(1,1,1,−3), (1,1,−3,1), (1,−3,1,1) and (−3,1,1,1),

the sum of the coordinates is zero and all coordinates are congruent to 1 (mod 4). Any three of these vectors generate the translation lattice.

The tessellations formed in this way from the order-2, order-3, and order-4 permutohedra, respectively, are the apeirogon, the regular hexagonal tiling, and the bitruncated cubic honeycomb. The dual tessellations contain all simplex facets, although they are not regular polytopes beyond order-3.

Examples

Order 2 Order 3 Order 4
2 vertices 6 vertices 24 vertices
The permutohedron of order 2 is a line segment on the diagonal of a unit square. The permutohedron of order 3 is a regular hexagon, filling a cross-section of a 2×2×2 cube. The permutohedron of order 4 forms a truncated octahedron.
Order 5 Order 6
120 vertices 720 vertices
The permutohedron of order 5 forms an omnitruncated 5-cell. The permutohedron of order 6 forms an omnitruncated 5-simplex.

See also

Wikimedia Commons has media related to Permutohedron.

Notes

  1. ↑ Thomas (2006).
  2. ↑ Original French: "le mot permutoèdre est barbare, mais il est facile à retenir; soumettons le aux critiques des lecteurs."
  3. ↑ Gaiha & Gupta (1977).
  4. ↑ See, e.g., Ziegler (1995), p. 18.
  5. ↑ Ziegler (1995), p. 200.
  6. ↑ This Cayley graph labeling is shown, e.g., by Ziegler (1995).

References

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