A051459 Number of orderings of the subsets of a set with n elements that are compatible with the subsets' sizes; i.e., if A, B are two subsets with A <= B then Card(A) <= Card(B).

1, 1, 2, 36, 414720, 189621927936000000, 2156695499113014719143826715127578624000000000000

Offset: 0

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 15 2003

Keywords

• nonn

Comments

a(7) has 127 digits and too large to include in sequence. - Ray Chandler, Nov 22 2003

From Valentin Bakoev, Nov 20 2017, May 17 2019: (Start)

a(n) is the number of possible orderings of the vectors of the n-dimensional Boolean cube (hypercube) {0,1}^n in accordance with their (Hamming) weights. For arbitrary vectors u, v of {0, 1}^n, if wt(u)

a(n) is also the number of all possible topological orders (sortings) of the directed acyclic graph (DAG) defined by the same poset: {0,1}^n and the relation weight order as it is defined and explained above.

Both comments correspond to the name of the sequence since the corresponding Boolean algebras are isomorphic. (End)

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..9
• Valentin Bakoev, Some problems and algorithms related to the weight order relation on the n-dimensional Boolean cube, Discrete Mathematics, Algorithms and Applications, Vol. 13 No 3, 2150021 (2021); arXiv preprint, arXiv:1811.04421 [cs.DM], 2018-2020.

Crossrefs

Cf. A000722, A001142, A022914, A294648.

Programs

Maple

a:= n-> mul(binomial(n, i)!, i=0..n):
seq(a(n), n=0..6);  # Alois P. Heinz, Nov 20 2017

Mathematica

Array[Product[Binomial[#, i]!, {i, #}] &, 7, 0] (* Michael De Vlieger, Nov 20 2017 *)

Maxima

a(n):= prod(binomial(n,k)!,k,0,n); /* Valentin Bakoev, May 17 2019 */

PARI

a(n) = prod(k=0, n, binomial(n, k)!); \\ Michel Marcus, May 18 2019

Formula

a(n) = C(n, 0)! * C(n, 1)! * C(n, 2)! * ... * C(n, n)! = A000722(n) / A022914(n).

log(a(n)) ~ log(2) * n * 2^n. - Vaclav Kotesovec, Nov 24 2023

Extensions

More terms from Ray Chandler, Nov 22 2003

a(0)=1 prepended by Alois P. Heinz, Nov 20 2017

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