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A083355
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Number of preferential arrangements for the set partitions of the n-set [1,2,3,...,n].
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10
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1, 1, 4, 23, 175, 1662, 18937, 251729, 3824282, 65361237, 1241218963, 25928015368, 590852003947, 14586471744301, 387798817072596, 11046531316503163, 335640299372252595, 10835556229612637150, 370383732831919278037
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Labeled analogue of A055887. See combstruct commands for more precise definition.
Stirling transform of A000670(n)=[1,3,13,75,...] is a(n)=[1,4,23,175,...]. - Michael Somos Mar 04 2004
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REFERENCES
| Thomas Wieder: The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [From Thomas Wieder (thomas.wieder(AT)t-online.de), Nov 14 2009]
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LINKS
| K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem [J. Phys. A 37 (2004), 3475-3487]
N. J. A. Sloane, Transforms
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
Thomas Wieder, Further comments on A083355
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FORMULA
| E.g.f.: 1/(2-exp(exp(x)-1)).
Representation as a double infinite series (Dobinski-type formula), in Maple notation: a(n)=sum(k^n/k!*sum(p^k/(2*exp(1))^p, p=1..infinity), k=1..infinity)/2, n=1, 2... From Karol A. Penson (penson(AT)lptl.jussieu.fr) and Pawel Blasiak (blasiak(AT)lptl.jussieu.fr), Nov 30 2003.
a(n) ~ n!/(2 * c * (log c)^(n+1)) where c = 1 + log 2.
a(n) = Sum_{k=1..n} binomial(n, k)*Bell(k)*a(n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 24 2003
a(n) = Sum_{i=1}^{n} Sum_{j=1}^{i} j!*stirling2(i, j)*stirling2(n, i) - Thomas Wieder (wieder.thomas(AT)t-online.de), May 09 2005
a(n)=sum_{k=1}^n S2(n,k) A000670(k)
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EXAMPLE
| Let the colon ":" be a separator between two levels. E.g. in {1,2}:{3} the set {1,2} is on the first level, the set {3} is on the second level.
n=2 gives A083355(2)=4 because we have {1,2} {1}{2} {1}:{2} {2}:{1}.
n=3 gives A083355(3)=23 because we have:
{1,2,3}
{1,2}{3} {1,2}:{3} {3}:{1,2}
{1,3}{2} {1,3}:{2} {2}:{1,3}
{2,3}{1} {2,3}:{1} {1}:{2,3}
{1}{2}{3}
{1}:{2}:{3}
{3}:{1}:{2}
{2}:{3}:{1}
{1}:{3}:{2}
{2}:{1}:{3}
{3}:{2}:{1}
{1}{2}:{3} {1}{3}:{2} {2}{3}:{1}
{1}:{2}{3} {2}:{1}{3} {3}:{1}{2}.
Examples for the unlabeled case A055887:
n=2 gives A055887(2)=3 because {1,1} {{1}:{1}} {2}
n=3 gives A055887(3)=8 because {1,1,1} {{1}:{1,1}} {{1,1}:{1}} {{1}:{1}:{1}} {1,2} {{1}:{2}} {{2}:{1}} {3}.
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MAPLE
| with(combstruct); SeqSetSetL := [T, {T=Sequence(S), S=Set(U, card >= 1), U=Set(Z, card >= 1)}, labeled]; A083355 := n-> count(SeqSetSetL, size=n);
A083355 := proc(n::integer) #with(combinat); local a, i, j; a:=0; for i from 1 to n do for j from 1 to i do a := a + j!*stirling2(i, j)*stirling2(n, i); od; od; print("n, a(n): ", n, a); end proc; (Wieder)
A083355 := proc() local a, k, n; for n from 1 to 12 do a[n]:=0: for k from 1 to n do a[n]:=a[n]+stirling2(n, k)*A000670(k): od: od: print(a[1], a[2], a[3], a[4], a[5], a[6], a[7], a[8], a[9], a[10], a[11], a[12]); end proc; A000670 := proc(n) local Result, k; Result:=0: for k from 1 to n do Result:=Result+stirling2(n, k)*k! od: end proc;
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MATHEMATICA
| Range[0, 18]!CoefficientList[Series[1/(2 - E^(E^x - 1)), {x, 0, 18}], x] (from Robert G. Wilson v Jul 13 2004)
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PROG
| (PARI) a(n)=if(n<0, 0, n!*polcoeff(1/(2-exp(exp(x+x*O(x^n))-1)), n))
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CROSSREFS
| Cf. A055887.
Cf. A000079, A000670, A055887, A034691, A075729.
Sequence in context: A084357 A075729 A127131 * A141763 A025550 A067545
Adjacent sequences: A083352 A083353 A083354 * A083356 A083357 A083358
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KEYWORD
| nonn
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AUTHOR
| Thomas Wieder (wieder.thomas(AT)t-online.de), Jun 11 2003, May 07 2008
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