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OFFSET
| 1,2
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COMMENTS
| Number of set partitions of all nonempty subsets of a set, Bell(2^n-1).
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FORMULA
| a(n) = sum((stirling2(2^n-1,k)), k=1..2^n-1) = bell(2^n-1), where stirling2(n) is the Stirling number of the second kind and Bell(n) is the Bell number.
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EXAMPLE
| Let S={1,2,3,...,n} be a set of n elements and let
SU be the set of all nonempty subsets of S. The number of elements
of SU is |SU| = 2^n-1. Now form all possible set partitions from SU
where the empty set is excluded. This gives a set W and its number of
elements is |W| = sum((stirling2(2^n-1,k)), k=1..2^n-1).
For S={1,2} we have SU = { {1}, {2}, {1,2} } and W =
{
{{1}, {2}, {1, 2}},
{{1, 2}, {{1}, {2}}},
{{2}, {{1}, {1, 2}}},
{{1}, {{2}, {1, 2}}},
{{{1}, {2}, {1, 2}}}
}
and |W| = 5.
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MAPLE
| ZahlDerMengenAusMengeDerZerlegungenEinerMenge:=proc() local n, nend, arg, k, w; nend:=5; for n from 1 to nend do arg:=2^n-1; w[n]:=sum((stirling2(arg, k)), k=1..arg); od; print(w[1], w[2], w[3], w[4], w[5], w[6], w[7], w[8], w[9], w[10]); end proc;
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CROSSREFS
| Cf. A000079, A000110, A008277, A077585, A135085.
Sequence in context: A198402 A085706 A190350 * A206356 A094630 A173914
Adjacent sequences: A135081 A135082 A135083 * A135085 A135086 A135087
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KEYWORD
| nonn
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AUTHOR
| Thomas Wieder (thomas.wieder(AT)t-online.de), Nov 18 2007
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