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 A099880 Number of preferential arrangements (or simple hierarchies) of 2*n labeled elements with two kinds of elements (where each kind has n elements). 2
 1, 2, 18, 260, 5250, 136332, 4327092, 162309576, 7024896450, 344582629820, 18890850749628, 1144656941236536, 75963981061424820, 5479642938171428600, 426894499408073653800, 35720957482170932284560, 3195135789350678836128450, 304234032845362459798904220 (list; graph; refs; listen; history; text; internal format)
 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: A143138 A151362 A215362 * 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

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Last modified October 28 19:51 EDT 2020. Contains 338064 sequences. (Running on oeis4.)