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

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

Offset: 1

Author

Victor S. Miller, Jan 10 2014

Keywords

• nonn
• hard
• more

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.

Examples

			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.
		

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

• 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.

Crossrefs

Cf. A053043, A246427.

Programs

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())