Talk:Random permutation statistics

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This article needs more context around or a better explanation of technical details to make it more accessible to general readers and technical readers outside the specialty, without removing technical details. See below for more information.


[edit] Article needs work

I read this article thorugh and found it unnecessary technical and lacking some structure. You might want to start with an introduction. What statistics are you considering? Why? Can your results be proved by other methods? Here are few more precise recommendations:

1) Give an outline of the article. Discuss the scope.
2) Move the "fundamental relation" to a separate WP page with this title. Refer to it if necessary. This part is about technique, not statistics and this should be understood by the reader.
3) In each paragraph start with the statement of the theorem. Include your proof at the end. Say what are other proofs.

Examples:

  • Derangement -- start with the formula upfront. Give the involution principle proof. Conclude with g.f. as a remark.
  • "Expected number of cycles" -- start with the formula. Give a proof using basic counting: expected numberof cycles of length m = {n \choose m} (m-1)! / n! = 1/m Only then g.f. proof can be used.
  • "Expected number of cycles of any length of a random permutation" -- from the previous result, obviously, it's the sum of 1/m for m=1..n. Only after you state and explain this you can incude you g.f. methods.
  • "Expected number of transpositions of a random permutation" -- misnamed. I have no idea what that means. Are you counting 2-cycles? Perhaps inversions? This is unclear.
  • "Expected cycle size of a random element" -- confusing. You really meant expected cycle size containing element 1, but wanted to emphasize the symmetry. Then SAY so! First state a theorem that the cycle containing 1 has uniform length. Prove it by simple counting. Conclude that the expected length in (n+1)/2. Only then you can enclose the g.f. proof.
  • "Expected number of inversions" -- this is unforgivingly complicated. A trivial bijective argument (a_1,a_2...,a_n) --> (a_n,...,a_2,a_1) gives the symmetry of the number of inversions around {n \choose 2}/2. The result follows. Same can be concluded from the explicit product formula. Are you sure you want to use g.f.?

In summary, the article clearly needs work. Mhym 09:48, 4 December 2006 (UTC)

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