Algorithms for enumerating the exact null distributions of Kendall's S and Spearman's D statistics, when there are ties in one or both of the rankings, are presented. An expression, which is used to provide a simple proof of the asymptotic normality of the score S when both rankings are tied, is obtained for the cumulant generating function of S. The usefulness of an Edgeworth approximation to the null distribution of S in the general case of tied rankings is investigated and compared with the standard normal approximation.;Exact and asymptotic results are developed for the distribution of Kendall's partial rank correlation statistic {dollar}t\sb{lcub}12.3{rcub}{dollar}, under the complete null hypothesis. A probability model, with the property that for the associated permutations E(t) = {dollar}\tau{dollar}, is developed for the elements of an inversion vector. The variance of t under this probability model is derived, an application of this result to hypothesis testing is presented, and an algorithm for simulating rankings of size n, so that E(t) = {dollar}\tau{dollar}, is given.;An asymptotic variance estimator for {dollar}t\sb{lcub}12.3{rcub}{dollar} is derived and the asymptotic normality of {dollar}t\sb{lcub}12.3{rcub}{dollar}, under {dollar}H\sb o{dollar}: {dollar}\tau\sb{lcub}12.3{rcub}{dollar} = 0 and for the general case of variates with underlying parental correlation, is established. Monte Carlo simulation is used to show that when the magnitudes of {dollar}t\sb{lcub}13{rcub}{dollar} and {dollar}t\sb{lcub}23{rcub}{dollar} are both moderately large, {dollar}t\sb{lcub}12.3{rcub}{dollar} is not a suitable statistic for testing the hypothesis {dollar}H\sbsp{lcub}o{rcub}{lcub}\prime{rcub}{dollar}: {dollar}X\sb1{dollar} and {dollar}X\sb2{dollar} are conditionally, given {dollar}X\sb3{dollar}, independent of each other. Consequently, a simulation study of partial Spearman's {dollar}\rho{dollar} is implemented. This study shows that {dollar}r\sb{lcub}s,12.3{rcub}{dollar}, when corrected for bias in {dollar}r\sb{lcub}s,12{rcub}{dollar} etc., provides a satisfactory test statistic whose asymptotic distribution under {dollar}H\sb o{dollar}: {dollar}\rho\sb{lcub}\rm s,12{rcub}{dollar} = 0 may be adequately approximated by its asymptotic distribution under the complete null hypothesis.