A248791 Decimal expansion of P_2(xi), the maximum limiting probability that a random n-permutation has exactly two cycles exceeding a given length.
0, 7, 2, 7, 8, 8, 7, 3, 8, 6, 6, 0, 8, 2, 1, 3, 7, 3, 3, 6, 6, 7, 1, 8, 6, 6, 3, 3, 1, 8, 1, 9, 1, 4, 2, 9, 6, 8, 8, 9, 2, 9, 5, 4, 9, 4, 4, 8, 7, 0, 6, 8, 4, 1, 4, 5, 7, 5, 1, 3, 1, 8, 3, 4, 6, 1, 4, 4, 6, 0, 6, 9, 1, 6, 6, 9, 0, 2, 2, 7, 6, 4, 0, 1, 7, 0, 8, 1, 9, 5, 9, 2, 9, 2, 0, 8, 3, 6, 2, 6, 9
Offset: 0
Examples
0.0727887386608213733667186633181914296889295494487...
Links
- Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 29.
- Michael Lugo, The number of cycles of specified normalized length in permutations, arXiv:0909.2909 [math.CO] 2009.
Programs
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Mathematica
xi = 1/(1 + Sqrt[E]); P2[x_] := -Pi^2/12 + (1/2)*Log[x]^2 + PolyLog[2, x]; Join[{0}, RealDigits[P2[xi], 10, 100] // First] -
Python
from mpmath import * mp.dps=101 xi=1/(1 + sqrt(e)) C = -pi**2/12 + (1/2)*log(xi)**2 + polylog(2, xi) print([int(n) for n in list(str(C)[2:-1])]) # Indranil Ghosh, Jul 03 2017
Formula
-Pi^2/12 + (1/2)*log(1 + sqrt(e))^2 + Li_2(1/(1 + sqrt(e))).