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A189232
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Triangle read by rows: Number of crossing set partitions of {1,2,...,n} into k blocks.
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2
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0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 5, 0, 0, 0, 16, 40, 15, 0, 0, 0, 42, 196, 175, 35, 0, 0, 0, 99, 770, 1211, 560, 70, 0, 0, 0, 219, 2689, 6594, 5187, 1470, 126, 0, 0, 0, 466, 8790, 31585, 37233, 17535, 3360, 210, 0, 0
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OFFSET
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1,12
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999 (Exericses 6.19)
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LINKS
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Table of n, a(n) for n=1..55.
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FORMULA
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T(n,k) = S2(n,k) - C(n,k-1)*C(n,k)/n; S2(n,k) Stirling numbers of the second kind, C(n,k) binomial coefficients.
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EXAMPLE
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There are 10 crossing set partitions of {1,2,3,4,5}.
T(5,2) = card{13|245, 14|235, 24|135, 25|134, 35|124} = 5.
T(5,3) = card{1|35|24, 2|14|35, 3|14|25, 4|13|25, 5|13|24} = 5.
[1] 0
[2] 0, 0
[3] 0, 0, 0
[4] 0, 1, 0, 0
[5] 0, 5, 5, 0, 0
[6] 0, 16, 40, 15, 0, 0
[7] 0, 42, 196, 175, 35, 0, 0
[8] 0, 99, 770, 1211, 560, 70, 0, 0
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MAPLE
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A189232 := (n, k) -> combinat[stirling2](n, k) - binomial(n, k-1)*binomial(n, k)/n:
for n from 1 to 9 do seq(A189232(n, k), k = 1..n) od;
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MATHEMATICA
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T[n_, k_] := StirlingS2[n, k] - Binomial[n, k-1] Binomial[n, k]/n;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* Jean-François Alcover, Jun 24 2019 *)
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CROSSREFS
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Row sums A016098, A001263.
Sequence in context: A144776 A065937 A197738 * A247667 A115144 A200506
Adjacent sequences: A189229 A189230 A189231 * A189233 A189234 A189235
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KEYWORD
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nonn,tabl
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AUTHOR
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Peter Luschny, Apr 28 2011
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STATUS
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approved
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