# Number of permutations without a cycle of length k

• Nov 24th 2011, 02:18 PM
boony
Number of permutations without a cycle of length k
Hello,

how many permutations in S_n are there without any cycle of length k?

boony
• Nov 24th 2011, 04:41 PM
Annatala
Re: Number of permutations without a cycle of length k
From Wikipedia, the unsigned Stirling number of the first kind counts the number of permutations of $e$ elements with exactly $c$ disjoint cycles:

Definition by recursion (defined only when $e$ $\geq$ $c$ $\geq$ $1$):
if $e = 1$: $s(e, 1) = (e-1)!$
if $e > 1$: $s(e, c) = s(e-1, c-1) + s(e-1, c) * (e-1)$

(That last one implies anytime $e = c,$ the answer is $1$.)

EDIT: This doesn't actually answer your question, I now notice. (Doh) But the Wikipedia page should help, I think. To figure out the number of permutations without cycles of length k, it's probably easiest to calculate the number of permutations that do have cycles of length k (or of length at least k, depending on how it's worded), and subtract that from the total number of permutations.