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A008827
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Number of proper partitions of a set of n labeled elements.
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5
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0, 3, 13, 50, 201, 875, 4138, 21145, 115973, 678568, 4213595, 27644435, 190899320, 1382958543, 10480142145, 82864869802, 682076806157, 5832742205055, 51724158235370, 474869816156749, 4506715738447321, 44152005855084344, 445958869294805287, 4638590332229999351
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OFFSET
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2,2
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COMMENTS
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Previous name: Coefficients from fractional iteration of exp(x) - 1.
A "proper partition" of a set is a set partition in which there is more than one part, and there is some part which has more than one element.
a(n) is the number of chains of length 2 from the top element to the bottom element in the partition lattice on n labeled objects.
(End)
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.
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LINKS
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FORMULA
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EXAMPLE
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For n = 3 there are a(3) = 3 proper partitions of {1,2,3}, which can be represented {12|3}, {13|2}, {23|1}.
For n = 4 there are a(4) = 13 proper partitions of {1,2,3,4}, which can be represented {123|4}, {124|3}, {134|2}, {234|1}, {12|34}, {13|24}, {14|23}, {12|3|4}, {13|2|4}, {14|2|3}, {23|1|4}, {24|1|3}, {34|1|2}.
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MAPLE
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MATHEMATICA
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PROG
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(Magma) [Bell(n) -2: n in [3..30]]; // G. C. Greubel, Sep 13 2019
(Sage) [bell_number(n)-2 for n in (3..30)] # G. C. Greubel, Sep 13 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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