Hello,

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

Thanks for the answers

boony

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- Nov 24th 2011, 02:18 PMboonyNumber of permutations without a cycle of length k
Hello,

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

Thanks for the answers

boony - Nov 24th 2011, 04:41 PMAnnatalaRe: 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 $\displaystyle e$ elements with exactly $\displaystyle c$ disjoint cycles:

Definition by recursion (defined only when $\displaystyle e$ $\displaystyle \geq$ $\displaystyle c$ $\displaystyle \geq$ $\displaystyle 1$):

if $\displaystyle e = 1$: $\displaystyle s(e, 1) = (e-1)!$

if $\displaystyle e > 1$: $\displaystyle s(e, c) = s(e-1, c-1) + s(e-1, c) * (e-1)$

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

**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.