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A122949
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Number of ordered pairs of permutations generating a transitive group.
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1
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1, 3, 26, 426, 11064, 413640, 20946960, 1377648720, 114078384000, 11611761920640, 1425189271161600, 207609729886944000, 35419018603306060800, 6996657393055480550400, 1584616114318716544665600
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| From Dixon: The sequence is asymptotic to (n!)^2; when divided by n!^2, it has a high-order asymptotic contact with the probability that two randomly chosen permutations generate the symmetric group. Also: a(n)=(n-1)!*A003319(n+1), where A003319 is the number of connected [or indecomposable] permutations. The coefficients in the asymptotic expansion of a(n)/(n!)^2 are A113869 and in absolute value, they constitute A084357 (number of sets of sets of lists).
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LINKS
| John D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, vol 11(2), R56.
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FORMULA
| Exponential generating function is: log(1+sum(n!*z^n,n=1..infinity))
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EXAMPLE
| a(2)=3 because there are 2!*2!=4 pairs of permutations, of which only [(1,1),(1,1)] does not generate a transitive group.
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MAPLE
| series(log(add(n!*z^n, n=0..Order+2)), z=0):seq(coeff(%, z, j)*j!, j=0..Order);
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MATHEMATICA
| max = 15; Drop[ CoefficientList[ Series[ Log[1 + Sum[n!*z^n, {n, 1, max}]], {z, 0, max}], z]* Range[0, max]!, 1](* From Jean-François Alcover, Oct 05 2011 *)
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CROSSREFS
| Cf. A003319, A084357, A113869.
Sequence in context: A206403 A192554 A206402 * A182958 A174423 A049088
Adjacent sequences: A122946 A122947 A122948 * A122950 A122951 A122952
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KEYWORD
| nonn
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AUTHOR
| Philippe.Flajolet(AT)inria.fr (Philippe.Flajolet(AT)inria.fr), Oct 25 2006
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