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 A235459 Number of facets of the correlation polytope of degree n. 1
 2, 4, 16, 56, 368, 116764, 217093472 (list; graph; refs; listen; history; text; internal format)
 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,...,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 1-41. M. M. Deza, and M. Laurent, Geometry of Cuts and Metrics, Springer, 1997, pp. 52-54 LINKS T. Christof, The SMAPO database about the CUT polytope G. Ziegler, Lectures on 0/1 Polytopes, arXiv:math/9909177v1 (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 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

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Last modified September 16 07:19 EDT 2021. Contains 347469 sequences. (Running on oeis4.)