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Search: id:A060725
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| A060725 |
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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 5-cycle. |
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3
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| 1, 1, 2, 6, 24, 96, 576, 4032, 32256, 290304, 2975616, 32731776, 392781312, 5106157056, 71486198784, 1070549415936, 17128790654976, 291189441134592, 5241409940422656, 99586788868030464, 1991897970827821056
(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^5)/5) / (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/5 ]( (-1)^i /(i! * 5^i)) by this formula we have as n -> infinity: a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 5^i) = e^(-1/5) or a(n) ~ e^(-1/5) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/5) * (n/e)^n * sqrt(2 * Pi * n)
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EXAMPLE
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a(5) = 96 because in S_5 the permutations with no 5-cycle are the complement of the 24 5-cycles so a(5) = 5! - 24 = 96.
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MAPLE
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for n from 0 to 30 do printf(`%d, `, n! * sum(( (-1)^i /(i! * 5^i)), i=0..floor(n/5))) od:
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CROSSREFS
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A000142.
Sequence in context: A070945 A053502 A053504 this_sequence A094012 A141253 A078486
Adjacent sequences: A060722 A060723 A060724 this_sequence A060726 A060727 A060728
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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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