Consider the random variable:
number of inversions (equivalently, major index)
of a (uniformly) drawn n-permutation
2
The mean is:, -1/4 n + 1/4 n
2 3
The variance is:, -5/72 n + 1/24 n + 1/36 n
The asympotics to order 1, of the even alpha coefficients (the (2r)-th momen\
t about the mean divided by the r-th power of the variance)
as an expression in n and r is:
/ 9 r (r - 1)\
(2 r)! |1 - -----------|
\ 25 n /
------------------------
r
r! 2
the (normalized) odd moments are
0
as expected, it being a symmetric prob. distibution
In particular it is (as first proved by Feller) asymptotically normal, but w\
e have an even finer asympotics for the alpha coefficients.
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