

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
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OFFSET

1,2


COMMENTS

From Dixon: The sequence is asymptotic to (n!)^2; when divided by n!^2, it has a highorder asymptotic contact with the probability that two randomly chosen permutations generate the symmetric group. Also: a(n)=(n1)!*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



FORMULA

Exponential generating function is: log(1+Sum_{n>=1}n!*z^n).
a(n) = (n!)^2  (n1)! * Sum_{k=1..n1} a(k) * (nk)! / (k1)!.  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](* JeanFrançois 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



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



