

A235459


Number of facets of the correlation polytope of degree n.


2




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. 5254.
G. Kalai and G. Ziegler, ed. "Polytopes: Combinatorics and Computation", Springer, 2000, Chapter 1, pp 141.


LINKS



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



KEYWORD

nonn,hard,more


AUTHOR



STATUS

approved



