This recent question makes me wonder: is there some known limit theorem for the distribution of the sizes of conjugacy classes in the symmetric group $S_n?$ A quick search seems to reveal nothing relevant.

You will maybe find some relevant information in the book "Logarithmic combinatorial structures: a probabilistic approach" by Richard Arratia,A. D. Barbour,Simon Tavaré.
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BS.Dec 26 '11 at 0:10

Probably you are already aware of the "arcsin law" (due to Vershik and Kerov) for the dual question about the limiting distribution of degrees of irreducible characters of S_n.
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John Wiltshire-GordonDec 26 '11 at 5:43

I deleted my answer pointing to work of Vershik, since it answers the question of a limiting distribution on partitions weighted by the size of corresponding conjugacy classes. I still think that one can get an answer to Igor's question from that. Things would be easier if the space of positive series which sum to 1 had a Lebesgue measure...
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Gjergji ZaimiDec 26 '11 at 10:41

@Gjergji Thanks! I know there is a lot of work on the general probability theory of the symmetric group. I will take a look at Vershik's talk...
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Igor RivinDec 26 '11 at 12:49

The distribution is going to spread out quite a bit. For instance, the largest conjugacy class has cycle lengths $(n-1,1)$ and is of size $n(n-2)! = n!/(n-1)$. The number of permutations whose longest cycle has length at least $n/2$ is asymptotic to $(\log 2)n!$.
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Richard StanleyDec 10 '12 at 20:57