A009997 -id:A009997 - cp's OEIS frontend

A005806 Number of comparative probability orderings on n elements.

1, 1, 1, 2, 14, 546, 169444, 560043206

Offset: 0

			For n = 3, the two orders are 1 < 2 < 12 < 3 < 13 < 23 < 123 and 1 < 2 < 3 < 12 < 13 < 23 < 123.
For zero elements, there is exactly one ordering. - _M. F. Hasler_, Mar 17 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Beveridge, Ian Calaway, and Kristin Heysse, de Finetti Lattices and Magog Triangles, arXiv:1912.12319 [math.CO], 2019.
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.

Cf. A009997.

a(n) >= A009997(n) with equality iff n < 5. - M. F. Hasler, Mar 17 2023

a(7) from Diane Maclagan and Michael Kleber

A361388 Number of orders of distances to vertices of n-dimensional cube.

Original entry on oeis.org

1, 2, 8, 96, 5376, 1981440, 5722536960, 138430238607360

Offset: 0

Pierre Abbat, Mar 10 2023

Let C be an n-dimensional cube and p be a point in R^n such that the distances from p to the 2^n vertices of C are all different. List the vertices in order of their distance from p. The number of different orders of vertices is given by a(n).
Equality of any two distances defines a hyperplane in R^n, although different pairs of distances may define the same hyperplane. All these hyperplanes partition the space into cells, and the interior of each (n-dimensional) cell corresponds to a particular strong order of the differences. Hence, a(n) equals the number of cells in the partition of R^n by the hyperplanes. The given SageMath code implements this approach. - Max Alekseyev, Mar 10 2023
Computing the sequence is slow. The Sage program took 20 minutes to compute a(5) on Lucas Brown's box; the C++ program took 3.5 seconds to compute a(5) on Pierre Abbat's box, a 12-thread Ryzen. The C++ program took 6 hours to compute a(6). Neither of us has computed a(7) with the program; that's from A009997.
For n >= 4 the frequencies of the orders appear to vary widely.

			For n=3, a 3-dimensional cube has 8 corners, numbered 0 to 7. A point can be closest to any of the 8 corners. A point closest to 0 can have distances to corners 1, 2, and 4 in any of 6 orders. A point whose distances to corners 0, 1, 2, and 4 are in increasing order can be closer to 3 than to 4, or closer to 4 than to 3. So the total number of orders is 8*6*2=96.

Pierre Abbat, Cubeorders

Cf. A009997.

A361388(n) = A009997(n)*n!<M. F. Hasler, Mar 10 2023

def a(n):
    x = polygens(QQ,n,'x')
    dist2 = [sum((xi - ti)^2 for xi,ti in zip(x,t)) for t in Tuples(range(2),n)]    # squared distances
    diffs = {p[0]-p[1] for p in Combinations(dist2,2)}     # set of pairwise differences of squared distances
    H = HyperplaneArrangements(QQ, tuple(map(str,x)))
    A = H([[[d.coefficient({xi:1}) for xi in x], d.constant_coefficient()] for d in diffs])
    return A.n_regions()
print( [a(n) for n in (1..4)] ) # Max Alekseyev, Mar 10 2023
(C++) // See Cubeorders link.

a(n) = 2^n*n!*A009997(n).

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A005806 Number of comparative probability orderings on n elements.

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A361388 Number of orders of distances to vertices of n-dimensional cube.

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