Consider the random variable:
number of inversions (equivalently, major index)
of all words in {1,2}* with exactly n 1's and exactly n 2's
2
n
The mean is:, ----
2
2 3
The variance is:, 1/12 n + 1/6 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:
/ 3 r (r - 1)\
(2 r)! |1 - -----------|
\ 10 n /
------------------------
r
r! 2
the (normalized) odd moments are
0
as expected, it being a symmetric prob. distibution
In particular it is asymptotically normal, but we have an even finer asympot\
ics for the alpha coefficients.
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