Leading Asympotitcs to order 3 of the moments of the Spearman's Footrule By Shalosh B. Ekhad Thereom: The first leading terms in the expression of the (2r)-th moment of \ Spearman's Footrule are / 2 r (3 r) | r (20 r - 501 r + 1363) 5 4 (2/45) (2 r)! n |1 + ------------------------ + r (57200 r - 8425560 r \ 882 n 3 2 / + 141955229 r - 528900858 r + 526923659 r + 2204496102) / (667458792 / \ 2 | / r n )| / (2 r!) / / and in Maple notation (2/45)^r*(2*r)!/(2^r)/r!*n^(3*r)*(1+1/882*r*(20*r^2-501*r+1363)/n+1/667458792*r *(57200*r^5-8425560*r^4+141955229*r^3-528900858*r^2+526923659*r+2204496102)/n^2 ) Also the first two terms in the exrpression for the (2r+1)-th moments are r (3 r + 1) (2/45) (2 r)! n / 3 2 \ | 2 r (2 r + 1) r (2 r + 1) (220 r - 15873 r + 39464 r + 63507)| / |- ------------- - -------------------------------------------------| / ( \ 63 916839 n / / r 2 r!) and in Maple notation (2/45)^r*(2*r)!/(2^r)/r!*n^(3*r+1)*(-2/63*r*(2*r+1)-1/916839*r*(2*r+1)*(220*r^3 -15873*r^2+39464*r+63507)/n) Note that this immediately implies asymptotic normality, as first proved by\ Persin Diaconis, in a joint paper with Ron Graham Spearman's Footrule as a measure of disarray, J. Royal Stat. Soc., Sectio\ n B, vol. 39, No. 2 (1977), pp. 262-268 . See Theorem 1 but gives much more!