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 A009997 Number of comparative probability orderings on all subsets of n elements that can arise by assigning a probability distribution to the individual elements. 1
 1, 1, 2, 14, 516, 124187, 214580603 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Also 1/(2^n*n!) * number of regions of hyperplane arrangements with normals (0,1,-1)^n. From David W. Wilson, Aug 15 2008: (Start) Also, number of possible orderings of the set of divisors of a product of n distinct primes. Let p1 < p2 < ... < p_n be primes (say p1 = p, p2 = q, p3 = r, ...) Consider the set M of divisors of p1*p2*...*p_n. How many ways can M be ordered? For n = 0, we have m = { 1 }, with 1 ordering. For n = 1, we have M = { 1, p }. There is 1 possible ordering, 1 < p. For n = 2, we have M = { 1, p, q, pq }. Remembering p < q, there is again 1 possible ordering, 1 < p < q < pq. For n = 3, we have M = { 1, p, q, r, pq, pr, qr, pqr }. There are 2 possible orderings here: 1 < p < q < r < pq < pr < qr < pqr, 1 < p < q < pq < r < pr < qr < pqr. (End) LINKS Antoine Deza, George Manoussakis, Shmuel Onn, Primitive Zonotopes, Discrete & Computational Geometry, 2017, p. 1-13. (See p. 5.) T. Fine and J. Gill, The enumeration of comparative probability relations, Ann. Prob. 4 (1976) 667-673. D. Maclagan, Boolean Term Orders and the Root System B_n, arXiv:math/9809134 [math.CO], 1998-1999. D. Maclagan, Boolean Term Orders and the Root System B_n, Order 15 (1999), 279-295. CROSSREFS Cf. A005806. Sequence in context: A160710 A271145 A277134 * A048137 A005806 A015184 Adjacent sequences:  A009994 A009995 A009996 * A009998 A009999 A010000 KEYWORD nonn,hard,nice AUTHOR EXTENSIONS a(6) and a(7) from Diane Maclagan and Michael Kleber Edited by N. J. A. Sloane, Nov 26 2008 STATUS approved

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