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Asymptotic variance (normalized by n^2) of the second longest cycle in a random permutation on n symbols.
2

%I #8 Apr 30 2016 23:16:59

%S 0,1,2,5,5,3,7,9,0,6,3,5,9,0,5,8,7,8,1,4,7,9,8,0,0,3,5,8,4,6,6,0,1,9,

%T 8,6,7,8,5,5,0,8,3,0,1,1,9,9,3,6,5,1,7,7,2,5,9,2,4,2,5,4,2,6,7,3,9,4,

%U 6,4,9,1,4,5,7,4,3,9,7,4,9,4,2,8,8,7,3,5,1,6,5,9,3,6,2,3,5,6,6

%N Asymptotic variance (normalized by n^2) of the second longest cycle in a random permutation on n symbols.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4 Golomb-Dickman Constant, p. 285.

%H Xavier Gourdon, <a href="http://algo.inria.fr/gourdon/thesis.html">Combinatoire, Algorithmique et Géométrie des Polynomes</a> Ecole Polytechnique, Paris 1996, page 152 [in French]

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Golomb-DickmanConstant.html">Golomb-Dickman Constant</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Golomb%E2%80%93Dickman_constant">Golomb-Dickman constant</a>

%F Integral_{0..infinity} x*(1 - exp(Ei(-x))*(1 - Ei(-x))) dx - (integral_{0..infinity} 1 - exp(Ei(-x))*(1 - Ei(-x)) dx)^2, where Ei is the exponential integral.

%e 0.012553790635905878147980035846601986785508301199365...

%t digits = 98; NIntegrate[x*(1 - E^ExpIntegralEi[-x]*(1 - ExpIntegralEi[-x]) ), {x, 0, 200}, WorkingPrecision -> digits + 5] - NIntegrate[1 - E^ExpIntegralEi[-x]*(1 - ExpIntegralEi[-x]), {x, 0, 200}, WorkingPrecision -> digits + 5]^2 // Join[{0}, RealDigits[#, 10, digits][[1]]]&

%Y Cf. A084945, A247398, A272413.

%K nonn,cons

%O 0,3

%A _Jean-François Alcover_, Apr 29 2016