Consider the random variable, studied by Richard Stanely:
the largest up-down subsequence in a random n-permutation.
2 n
The mean is:, 1/6 + ---
3
13 8 n
The variance is:, - --- + ---
180 45
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:
/ r (r - 1) (10 r - 713)\
(2 r)! |1 + ----------------------|
\ 1764 n /
-----------------------------------
r
r! 2
the (normalized) odd moments, to order 1, are
1/2 1/2
-1/252 (2 r)! 180 2
/ 3 2 \
| (r - 1) (1760 r - 381744 r + 1430752 r + 150351)| / r
|r - 1 + --------------------------------------------------| / (r! 2
\ 931392 n / /
1/2
n )
In particular it is asymptotically normal (as proved by Stanely), but we ha\
ve an even finer asympotics for the alpha coefficients.
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