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. This took , 434.715, seconds .