login
For n >= 1 a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 7-cycle.
5

%I #17 Feb 13 2018 17:58:49

%S 1,1,2,6,24,120,720,4320,34560,311040,3110400,34214400,410572800,

%T 5337446400,75613824000,1134207360000,18147317760000,308504401920000,

%U 5553079234560000,105508505456640000,2110170109132800000,44288746761093120000,974352428744048640000

%N For n >= 1 a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 7-cycle.

%C This is the expansion of exp ((-x^7)/7)/(1-x).

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

%H Harry J. Smith, <a href="/A060727/b060727.txt">Table of n, a(n) for n = 0..100</a>

%F The formula for a(n) is: a(n) = n! * sum i=0 ... [ n/7 ]( (-1)^i /(i! * 7^i)) by this formula we have as n -> infinity: a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 7^i) = e^(-1/7) or a(n) ~ e^(-1/7) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/7) * (n/e)^n * sqrt(2 * Pi * n)

%F 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=7, n>=0. - _Simon Plouffe_, Feb 18 2011

%e a(7) = 4320 because in S_7 the permutations with no 7-cycle are the complement of the 720 7-cycles so a(7) = 7! - 720 = 4320.

%p for n from 0 to 30 do printf(`%d,`, n! * sum(( (-1)^i /(i! * 7^i)), i=0..floor(n/7))) od:

%o (PARI) { for (n=0, 100, write("b060727.txt", n, " ", n! * sum(i=0, n\7, (-1)^i / (i! * 7^i))); ) } \\ _Harry J. Smith_, Jul 10 2009

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

%K nonn

%O 0,3

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

%E More terms from _James A. Sellers_, Apr 24 2001