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A122949 Number of ordered pairs of permutations generating a transitive group. 6
1, 3, 26, 426, 11064, 413640, 20946960, 1377648720, 114078384000, 11611761920640, 1425189271161600, 207609729886944000, 35419018603306060800, 6996657393055480550400, 1584616114318716544665600, 407930516160959891683584000, 118458533875304716189544448000 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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).
LINKS
John D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, vol 11(2), R56.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 139.
FORMULA
Exponential generating function is: log(1+Sum_{n>=1}n!*z^n).
a(n) = (n!)^2 - (n-1)! * Sum_{k=1..n-1} a(k) * (n-k)! / (k-1)!. - Ilya Gutkovskiy, Jul 10 2020
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.
MAPLE
series(log(add(n!*z^n, n=0..Order+2)), z=0):seq(coeff(%, z, j)*j!, j=0..Order);
MATHEMATICA
max = 15; Drop[ CoefficientList[ Series[ Log[1 + Sum[n!*z^n, {n, 1, max}]], {z, 0, max}], z]* Range[0, max]!, 1](* Jean-François Alcover, Oct 05 2011 *)
PROG
(PARI) N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, k!*x^k)))) \\ Seiichi Manyama, Mar 01 2019
CROSSREFS
Sequence in context: A305144 A206402 A300954 * A305113 A251664 A182958
KEYWORD
nonn
AUTHOR
Philippe Flajolet, Oct 25 2006
EXTENSIONS
More terms from Seiichi Manyama, Mar 01 2019
STATUS
approved

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Last modified May 26 08:37 EDT 2024. Contains 372815 sequences. (Running on oeis4.)