A099880 Number of preferential arrangements (or simple hierarchies) of 2*n labeled elements with two kinds of elements (where each kind has n elements).

1, 2, 18, 260, 5250, 136332, 4327092, 162309576, 7024896450, 344582629820, 18890850749628, 1144656941236536, 75963981061424820, 5479642938171428600, 426894499408073653800, 35720957482170932284560, 3195135789350678836128450, 304234032845362459798904220

Offset

0,2

Comments

The unlabeled case seems to be given by A003480, which can be generated by the following combstruct command: SeqUnionU := [S, {S=Sequence(Set(U,card>=1), card>=1), U=Union(a,b), a=Atom, b=Atom},unlabeled]; [seq(count(SeqUnionU, size=n), n=0..20)]; .

Links

Alois P. Heinz, Table of n, a(n) for n = 0..348

Formula

a(n) = binomial(2*n, n) * Sum_{k=0..n} k! * Stirling2(n, k).

a(n) = binomial(2*n, n) * A000670(n).

a(n) = A154921(2n,n). - Mats Granvik, Feb 07 2009

Example

Let a[1], a[2],...,a[n] and b[1],b[2],...,b[n] denote two kinds "a" and "b" of labeled elements where each kind as n elements in total.
Let ":" denote a level, e.g., if the elements a[1] and a[2] are on level L=1 and the element b[1] is on level L=2 then a[1]a[2]:b[1] is a preferrential arrangement (a simple hierarchy) with two levels.
Then for n=2 we have a(2) = 18 arrangements: a[1]a[2]; a[1]:a[2]; a[2]:a[1]; a[1]b[1]; a[1]:b[1]; b[1]:a[1]; a[1]b[2]; a[1]:b[2]; b[2]:a[1]; a[2]b[1]; a[2]:b[1]; b[1]:a[2]; a[2]b[2]; a[2]:b[2]; b[2]:a[2]; b[1]b[2]; b[1]:b[2]; b[2]:b[1].

Maple

a:=n-> add(binomial(2*n, n)*(Stirling2(n, k))*k!, k=0..n): seq(a(n), n=0..16); # Zerinvary Lajos, Oct 19 2006
# second Maple program:
b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=1..n)) end:
a:= n-> b(n)*(2*n)!/n!:
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 03 2019

Mathematica

f[n_] := Sum[l! StirlingS2[n, l] Binomial[2n, n], {l, n}]; Table[ f[n], {n, 0, 16}] (* Robert G. Wilson v, Nov 04 2004 *)

Crossrefs

Cf. A000670, A075729, A003480, A154921.

Sequence in context: A215362 A360974 A350461 * A141009 A143154 A032037

Adjacent sequences: A099877 A099878 A099879 * A099881 A099882 A099883

Keyword

nonn

Author

Thomas Wieder, Nov 02 2004

Extensions

More terms from Robert G. Wilson v, Nov 04 2004

a(0) corrected and edited by Alois P. Heinz, Feb 03 2019

Status

approved