From Wikipedia, the unsigned Stirling number of the first kind counts the number of permutations of elements with exactly disjoint cycles:

Definition by recursion (defined only when ):

if :

if :

(That last one implies anytime the answer is .)

EDIT: This doesn't actually answer your question, I now notice. 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.