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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). 6
1, 1, 2, 36, 414720, 189621927936000000, 2156695499113014719143826715127578624000000000000 (list; graph; refs; listen; history; text; internal format)
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)<wt(v) then u precedes v in this ordering. If wt(u)=wt(v) there is no precedence by weight between them and each of them can be before the other. The formula for a(n) is explained easily by the notion "layer" of the cube - the set of all vectors of equal weights. The k-th layer of the cube is formed by all vectors of weight k. Obviously, any vector from the i-th layer precedes any vector from the j-th layer, for 0 <= i < j <= n. Hence the vectors can be rearranged only into the same layer. The k-th layer of the cube consists of C(n,k) vectors of weight k, that can be rearranged in C(n,k)! ways, for k= 0..n. Finally, the formula is obtained by applying the multiplication rule.
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)
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.
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
a:= n-> mul(binomial(n, i)!, i=0..n):
seq(a(n), n=0..6); # Alois P. Heinz, Nov 20 2017
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
Sequence in context: A306644 A283261 A280420 * A358972 A073581 A078081
Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 15 2003
More terms from Ray Chandler, Nov 22 2003
a(0)=1 prepended by Alois P. Heinz, Nov 20 2017

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