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A235459 Number of facets of the correlation polytope of degree n. 1


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

%N Number of facets of the correlation polytope of degree n.

%C 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.

%C 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.

%C According to the SMAPO database it is conjectured that

%C 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.

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

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

%H T. Christof, <a href="http://www.iwr.uni-heidelberg.de/groups/comopt/software/SMAPO/cut/cut.html">The SMAPO database about the CUT polytope</a>

%H G. Ziegler, <a href="http://arxiv.org/abs/math/9909177">Lectures on 0/1 Polytopes</a>, arXiv:math/9909177v1 (1999), p 22-28.

%e 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.

%o (sage)

%o def Correlation(n):

%o if n == 0:

%o yield (tuple([]),tuple([]))

%o return

%o for x,y in Correlation(n-1):

%o yield (x + (0,),y + (n-1)*(0,))

%o yield (x + (1,),y + x)

%o def CorrelationPolytope(n):

%o return Polyhedron(vertices=[x + y for x,y in Correlation(n)])

%o def a(n):

%o return len(CorrelationPolytope(n).Hrepresentation())

%K nonn,hard,more

%O 1,1

%A _Victor S. Miller_, Jan 10 2014

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