

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).


4




OFFSET

0,3


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 ndimensional 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 kth layer of the cube is formed by all vectors of weight k. Obviously, any vector from the ith layer precedes any vector from the jth layer, for 0 <= i < j <= n. Hence the vectors can be rearranged only into the same layer. The kth 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)


LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..9
Valentin Bakoev, About the ordinances of the vectors of the ndimensional Boolean cube in accordance with their weights, arXiv:1811.04421 [cs.DM], 2018.
Valentin Bakoev, Fast Computing the Algebraic Degree of Boolean Functions, arXiv:1905.08649 [cs.DM], 2019.


FORMULA

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


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 *)


PROG

(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


CROSSREFS

Cf. A000722, A001142, A022914, A294648.
Sequence in context: A306644 A283261 A280420 * A073581 A078081 A077772
Adjacent sequences: A051456 A051457 A051458 * A051460 A051461 A051462


KEYWORD

nonn


AUTHOR

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


EXTENSIONS

More terms from Ray Chandler, Nov 22 2003
a(0)=1 prepended by Alois P. Heinz, Nov 20 2017


STATUS

approved



