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268:"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.
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Passages in this article are identical to passages in the linked paper "The statistics of random permutations." For example, the first paragraph is verbatim as are many other passages that I checked. So either the wikipedia article is copied from the paper without proper attribution or the paper is
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The article seems to use only one specialized and rather obscure method to solve problems. As such, it is not a good encyclopedia article. I hope someone who knows permutation statistics can rewrite the article from the ground up so it is more accessible and shows the methods most people in
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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:
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This article contains a lot of original research previously not covered in the literature. This doesn't mean that the content is wrong, it's just not backed up by reliable sources. Note that math forum contributions are not reliable sources.
365:. However, there are lots of other methods for solving these problems, and the article should mention those, and cite the literature (including the books I just mentioned). I'll try to improve the article a bit myself.
258:"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.
274:(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.?
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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.
263:"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.
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273:"Expected number of inversions" -- this is unforgivingly complicated. A trivial bijective argument (a_1,a_2...,a_n) --: -->
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3) In each paragraph start with the statement of the theorem. Include your proof at the end. Say what are other proofs.
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Moreover, these enumerations of permutations with special properties have nothing or little to do with "random". --
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The paper is dated June 8, 2006. Portions of the wikipedia article were posted prior to that date.
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The method is not "obscure"; a lot has been written about it - e.g. Flajolet and
Sedgewick's book
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Knowledge. If you would like to participate, please visit the project page, where you can join
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253:= {n \choose m} (m-1)! / n! = 1/m Only then g.f. proof can be used.
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