A064852 Number of orbits in A002619 consisting of n permutations.

1, 0, 0, 1, 4, 18, 102, 624, 4476, 36248, 329890, 3326054, 36846276, 444783906, 5811885808, 81729607680, 1230752346352, 19760412095328, 336967037143578, 6082255011151724, 115852476579789984, 2322315553090615850, 48869596859895986086, 1077167364116800207968

Offset

1,5

Comments

Also, the number of aperiodic oriented cycles on n nodes up to rotation of the nodes. See A324513 for illustrations of aperiodic undirected cycles. - Andrew Howroyd, Aug 16 2019

Links

Harry J. Smith, Table of n, a(n) for n = 1..100

Formula

a(n) = Sum_{k|n} mu(n/k)*phi(n/k)*(n/k)^k*k!/n^2 = A047918(n, n)/n^2.

Example

n=6: The orbit {(124635)(235146)(346251)(451362)(562413)(613524)} consists of 6 single permutations.

Mathematica

a[n_] := Sum[ MoebiusMu[n/k] * EulerPhi[n/k] * (n/k)^k * (k!/n^2), {k, Divisors[n]}]; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Jun 26 2012, after PARI *)

Prog

(PARI) for(n=1, 23, print(sumdiv(n, d, moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n^2)))
(PARI) { for (n=1, 100, a=sumdiv(n, d, moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n^2); write("b064852.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 28 2009

Crossrefs

Cf. A000010, A002619, A008683, A047918, A324513.

Sequence in context: A201826 A327833 A350267 * A229286 A191365 A335459

Adjacent sequences: A064849 A064850 A064851 * A064853 A064854 A064855

Keyword

nice,nonn

Author

Michael Steyer (m.steyer(AT)osram.de), Oct 06 2001

Extensions

Corrected and extended by Jason Earls and Vladeta Jovovic, Oct 08 2001

Status

approved