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

%I #26 Feb 03 2019 09:13:24

%S 1,2,18,260,5250,136332,4327092,162309576,7024896450,344582629820,

%T 18890850749628,1144656941236536,75963981061424820,

%U 5479642938171428600,426894499408073653800,35720957482170932284560,3195135789350678836128450,304234032845362459798904220

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

%C 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)]; .

%H Alois P. Heinz, <a href="/A099880/b099880.txt">Table of n, a(n) for n = 0..348</a>

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

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

%F a(n) = A154921(2n,n). - _Mats Granvik_, Feb 07 2009

%e 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.

%e 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.

%e 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].

%p 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

%p # second Maple program:

%p b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=1..n)) end:

%p a:= n-> b(n)*(2*n)!/n!:

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Feb 03 2019

%t 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 *)

%Y Cf. A000670, A075729, A003480, A154921.

%K nonn

%O 0,2

%A _Thomas Wieder_, Nov 02 2004

%E More terms from _Robert G. Wilson v_, Nov 04 2004

%E a(0) corrected and edited by _Alois P. Heinz_, Feb 03 2019