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. This took , 5.640, seconds .