7
$\begingroup$

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.

$\endgroup$
5
  • 1
    $\begingroup$ You will maybe find some relevant information in the book "Logarithmic combinatorial structures: a probabilistic approach" by Richard Arratia,A. D. Barbour,Simon Tavaré. $\endgroup$
    – BS.
    Dec 26, 2011 at 0:10
  • $\begingroup$ 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. $\endgroup$ Dec 26, 2011 at 5:43
  • $\begingroup$ 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... $\endgroup$ Dec 26, 2011 at 10:41
  • $\begingroup$ @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... $\endgroup$
    – Igor Rivin
    Dec 26, 2011 at 12:49
  • 2
    $\begingroup$ 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 $(\log 2)n!$. $\endgroup$ Dec 10, 2012 at 20:57

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.