A248080 Decimal expansion of P_0(xi), the maximum limiting probability that a random n-permutation has no cycle exceeding a given length.

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

Offset

0,2

Links

Table of n, a(n) for n=0..101.

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.

Formula

(1/2)*log(1 + sqrt(e))^2 - log(1 + sqrt(e)) + Li_2(1/(1 + sqrt(e))) - Pi^2/12 + 1.

Example

0.098711754480714692493721308237020679933333354780844...

Mathematica

xi = 1/(1 + Sqrt[E]); P0[x_] := Log[x]^2/2 + Log[x] + PolyLog[2, x] - Pi^2/12 + 1; Join[{0}, RealDigits[P0[xi], 10, 101] // First]

Prog

(Python)
from mpmath import *
mp.dps=102
xi=1/(1 + sqrt(e))
C = log(xi)**2/2 + log(xi) + polylog(2, xi) - pi**2/12 + 1
print([int(n) for n in list(str(C)[2:-1])]) # Indranil Ghosh, Jul 04 2017

Crossrefs

Cf. A143301, A246849.

Sequence in context: A129269 A094145 A002388 * A278828 A334448 A011116

Adjacent sequences: A248077 A248078 A248079 * A248081 A248082 A248083

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Oct 14 2014

Status

approved