

A235459


Number of facets of the correlation polytope of degree n.


1




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) = 12,246,651,158,320. This database also says that the above value of a(7) is conjectural, but Ziegler lists it as known.


REFERENCES

G. Kalai and G. Ziegler, ed. "Polytopes: Combinatorics and Computation", Springer, 2000, Chapter 1, pp 141.
M. M. Deza, and M. Laurent, Geometry of Cuts and Metrics, Springer, 1997, pp. 5254


LINKS

Table of n, a(n) for n=1..7.
T. Christof, The SMAPO database about the CUT polytope
G. Ziegler, Lectures on 0/1 Polytopes, arXiv:math/9909177v1 (1999), p 2228


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(n1):
yield (x + (0, ), y + (n1)*(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

Sequence in context: A009624 A009161 A009290 * A081919 A232664 A153954
Adjacent sequences: A235456 A235457 A235458 * A235460 A235461 A235462


KEYWORD

nonn,hard,more


AUTHOR

Victor S. Miller, Jan 10 2014


STATUS

approved



