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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice

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, and 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.
• Shane Kepley, Konstantin Mischaikow, and Lun Zhang, Computing linear extensions for Boolean lattices with algebraic constraints, arXiv:2006.02622 [math.CO], 2020.
• D. Maclagan, Boolean Term Orders and the Root System B_n, arXiv:math/9809134 [math.CO], 1998-1999. (Data in table on p.13)
• D. Maclagan, Boolean Term Orders and the Root System B_n, Order 15 (1999), 279-295. (Data in first table on p. 293.)

Crossrefs

Cf. A005806, A361388.

Programs

PARI

apply( {A009997(n)=if(n>4, [516, 124187, 214580603][n-4], (n-=!!n)^n\/2)}, [0..7]) \\ M. F. Hasler, Mar 17 2023

Formula

a(n) <= A005806(n) with equality iff n <= 4. - M. F. Hasler, Mar 17 2023

Extensions

a(6) and a(7) from Diane Maclagan and Michael Kleber

Edited by N. J. A. Sloane, Nov 26 2008

a(0) = 1 inserted by M. F. Hasler, Mar 17 2023