A235459 Number of facets of the correlation polytope of degree n.

2, 4, 16, 56, 368, 116764, 217093472

Offset

1,1

Comments

The correlation polytope of degree n is the set of symmetric n X n matrices, P such that P[i,j] = Prob(X[i] = 1 and X[j] = 1) where (X[1],...,X[n]) is a sequence of 0/1 valued random variables (not necessarily independent). It is the convex hull of all n X n symmetric 0/1 matrices of rank 1.

The correlation polytope COR(n) is affinely equivalent to CUT(n+1), where CUT(n) is the cut polytope of complete graph on n vertices -- the convex hull of indicator vectors of a cut delta(S) -- where S is a subset of the vertices. The cut delta(S) is the set of edges with one end point in S and one endpoint not in S.

According to the SMAPO database it is conjectured that a(8) = 12246651158320. This database also says that the above value of a(7) is conjectural, but Ziegler lists it as known.

References

M. M. Deza and M. Laurent, Geometry of Cuts and Metrics, Springer, 1997, pp. 52-54.

G. Kalai and G. Ziegler, ed. "Polytopes: Combinatorics and Computation", Springer, 2000, Chapter 1, pp 1-41.

Links

Table of n, a(n) for n=1..7.

T. Christof, The SMAPO database about the CUT polytope

Michel Deza and Mathieu Dutour Sikirić, Enumeration of the facets of cut polytopes over some highly symmetric graphs, Intl. Trans. in Op. Res., 23 (2016), 853-860; arXiv:1501.05407 [math.CO], 2015. [Confirms the value of a(7).]

Stefan Forcey, Encyclopedia of Combinatorial Polytope Sequences: Cut Polytope.

G. Ziegler, Lectures on 0/1 Polytopes, arXiv:math/9909177 [math.CO], 1999, p 22-28.

Example

a(2) corresponds to 0 <= p[1,2] <= p[1,1],p[2,2] and p[1,1] + p[2,2] - p[1,2] <= 1.

Prog

(Sage)
def Correlation(n):
   if n == 0:
      yield (tuple([]), tuple([]))
      return
   for x, y in Correlation(n-1):
      yield (x + (0, ), y + (n-1)*(0, ))
      yield (x + (1, ), y + x)
def CorrelationPolytope(n):
   return Polyhedron(vertices=[x + y for x, y in Correlation(n)])
def a(n):
   return len(CorrelationPolytope(n).Hrepresentation())

Crossrefs

Cf. A053043, A246427.

Sequence in context: A009624 A009161 A009290 * A081919 A362524 A232664

Adjacent sequences: A235456 A235457 A235458 * A235460 A235461 A235462

Keyword

nonn,hard,more

Author

Victor S. Miller, Jan 10 2014

Status

approved