read HISTABRUT:
print(`Consider the random variable: `):
print(`number of inversions (equivalently, major index)`):
print(`of a (uniformly) drawn n-permutation `):
gu:= Yofseq([seq(qfac(i,q),i=1..26)],q,12,n,20,1,r):
print(`The mean is:`, gu[1]):
print(`The variance is:` , gu[2]):
print(`The asympotics to order 1, of the even alpha coefficients (the (2r)-th moment about the mean divided by the r-th power of the variance) `):
print(`as an expression in n and r is:`):
print(gu[3]):
print(`the (normalized) odd moments are`):
print(gu[4]):
print(`as expected, it being a symmetric prob. distibution `):
print(``):
print(`In particular it is (as first proved by Feller) asymptotically normal, but we have an even finer asympotics for the alpha coefficients. `):
print(`This took `, time(), `seconds . `):
quit: