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A052889
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Number of rooted set partitions.
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11
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0, 1, 2, 6, 20, 75, 312, 1421, 7016, 37260, 211470, 1275725, 8142840, 54776761, 387022118, 2863489830, 22127336720, 178162416499, 1491567656472, 12959459317021, 116654844101140
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Total number of blocks of size one in all set partitions of set {1..n}. - Wouter Meeussen, Jul 28, 2003
With offset 1, number of permutations beginning with 12 and avoiding 12-3.
a(n) = number of partitions of {1...n+1} containing exactly one pair of consecutive integers, counted within a block. With offset t-1, number of partitions of {1...N} containing one string of t consecutive integers, where N=n+j, t=2+j, j = 0,1,2,.... - A. O. Munagi (amunagi(AT)yahoo.com), Apr 10 2005
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REFERENCES
| A. O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463.
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 865
S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns.
A. O. Munagi, Set Partitions with Successions and Separations,IJMMS 2005:3 (2005),451-463.
S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).
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FORMULA
| E.g.f.: exp(exp(x)-1)*x
a(n) = n*Bell(n-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 14 2003
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i<=j), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=(-1)^(n-1)*coeff(charpoly(A,x),x). [From Milan R. Janjic (agnus(AT)blic.net), Jul 08 2010]
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EXAMPLE
| a(3) = 6 because the partitions of {1, 2, 3, 4} containing a pair of consecutive integers are 124/3, 134/2, 14/23, 12/3/4, 1/23/4, 1/2/34.
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MAPLE
| spec := [S, {B=Set(C), C=Set(Z, 1 <= card), S=Prod(Z, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
Explanation of above combstruct commands using generating functions, from Mitch Harris, Jul 28 2003:
Z = an atom (each atom used is labeled), gf: Z(x) = x
C = Set(Z, card <= 1) is the set of positive integers; gf: C(x) = e^(Z(x)) - 1 = e^x - 1 (the -1 removes the empty set); [x^n]C = 1 means there is exactly one set with n atoms since each atom is labeled
B = Set(C) the set of (ordered) sets of integers = ordered set partitions; gf: B(x) = e^C(x) = e^(e^x - 1)
S = Prod(Z, B) pairs of an atom (Z) and an ordered set partition = an ordered set partition with an adjoining single atom. The adjoining atom corresponds to choosing a "root" in the partition; gf: S(x) = x B(x) = x*e^(e^x-1)
A052889 := n -> `if`(n=0, 0, n*combinat[bell](n-1)):
seq(A052889(n), n=0..20); # - Peter Luschny, Apr 19 2011
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MATHEMATICA
| a=Exp[x]-1; Range[0, 20]! CoefficientList[Series[ x Exp[a], {x, 0, 20}], x] (*Geoffrey Critzer, Nov 25 2011*)
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CROSSREFS
| Second column of triangle A033306.
Sequence in context: A150168 A145870 A134957 * A150169 A083691 A150170
Adjacent sequences: A052886 A052887 A052888 * A052890 A052891 A052892
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KEYWORD
| easy,nonn
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AUTHOR
| encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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