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Search: id:A060726
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| A060726 |
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For n >= 1 a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 6-cycle. |
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3
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| 1, 1, 2, 6, 24, 120, 600, 4200, 33600, 302400, 3024000, 33264000, 405820800, 5275670400, 73859385600, 1107890784000, 17726252544000, 301346293248000, 5419293175296000, 102966570330624000, 2059331406612480000
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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This is the expansion of exp ((-x^6)/6) /(1-x).
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.
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FORMULA
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The formula for a(n) is: a(n) = n! * sum i=0 ... [ n/6 ]( (-1)^i /(i! * 6^i)) by this formula we have as n -> infinity: a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 6^i) = e^(-1/6) or a(n) ~ e^(-1/6) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/6) * (n/e)^n * sqrt(2 * Pi * n)
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EXAMPLE
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a(6) = 600 because in S_6 the permutations with no 6-cycle are the complement of the 120 6-cycles so a(6) = 6! - 120 = 600.
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MAPLE
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for n from 0 to 30 do printf(`%d, `, n! * sum(( (-1)^i /(i! * 6^i)), i=0..floor(n/6))) od:
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CROSSREFS
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A000142.
Sequence in context: A062348 A072856 A070946 this_sequence A138619 A068200 A121987
Adjacent sequences: A060723 A060724 A060725 this_sequence A060727 A060728 A060729
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KEYWORD
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nonn
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AUTHOR
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Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 24 2001
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