  This paper develops an analogy between the cycle structure of, on the one
hand, random permutations with cycle lengths restricted to lie in an infinite
set $S$ with asymptotic density $\sigma$ and, on the other hand, permutations
selected according to the Ewens distribution with parameter $\sigma$. In
particular we show that the asymptotic expected number of cycles of random
permutations of $[n]$ with all cycles even, with all cycles odd, and chosen
from the Ewens distribution with parameter 1/2 are all ${1 \over 2} \log n +
O(1)$, and the variance is of the same order. Furthermore, we show that in
permutations of $[n]$ chosen from the Ewens distribution with parameter
$\sigma$, the probability of a random element being in a cycle longer than
$\gamma n$ approaches $(1-\gamma)^\sigma$ for large $n$. The same limit law
holds for permutations with cycles carrying multiplicative weights with average
$\sigma$. We draw parallels between the Ewens distribution and the
asymptotic-density case and explain why these parallels should exist using
permutations drawn from weighted Boltzmann distributions.
