A060726 For n >= 1, a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 6-cycle.

1, 1, 2, 6, 24, 120, 600, 4200, 33600, 302400, 3024000, 33264000, 405820800, 5275670400, 73859385600, 1107890784000, 17726252544000, 301346293248000, 5419293175296000, 102966570330624000, 2059331406612480000

Offset: 0

Author

Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001

Keywords

• nonn

Comments

This is the expansion of exp ((-x^6)/6) /(1-x).

Examples

			a(6) = 600 because in S_6 the permutations with no 6-cycle are the complement of the 120 6-cycles so a(6) = 6! - 120 = 600.
		

References

• R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..100
• Plouffe, Simon, Exact formulas for integer sequences

Crossrefs

Cf. A000142, A000266, A000090, A000138, A060725, A060726, A060727.

Programs

Maple

for n from 0 to 30 do printf(`%d,`, n! * sum(( (-1)^i /(i! * 6^i)), i=0..floor(n/6))) od:

PARI

a(n)={n! * sum(i=0, n\6, (-1)^i / (i! * 6^i))} \\ Harry J. Smith, Jul 10 2009

Formula

The formula for a(n) is: a(n) = n! * Sum_{i=0..floor(n/6)} ((-1)^i /(i! * 6^i)) by this formula we have as n -> infinity: a(n)/n! ~ Sum_{i>= 0} (-1)^i /(i! * 6^i) = e^(-1/6) or a(n) ~ e^(-1/6) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/6) * (n/e)^n * sqrt(2 * Pi * n)

a(n,k) = n!*floor(floor(n/k)!*k^floor(n/k)/exp(1/k) + 1/2)/(floor(n/k)!*k^floor(n/k)), k=6, n >= 0. - Simon Plouffe, Feb 18 2011

Extensions

More terms from James Sellers, Apr 24 2001