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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

Table of n, a(n) for n=1..7.

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

N. J. A. Sloane

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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Last modified November 23 21:46 EST 2017. Contains 295141 sequences.