The relation between Mann–Whitney's statistic and Kendall's correlation coefficient $\tau$

It is shown that Kendall's correlation coefficient may be expressed as follows:
$$ \tau=\frac{4\sum_{s=1}^k\sum_{f=1}^{s-1}U_{sf}-N^2+\sum_{s=1}^kn_{s\cdot}^2}{\sqrt{(N^2-\sum n_{s\cdot}^2)(N^2-\sum n_{\cdot t}^2)}} $$
where $N$ is the sample size $U_sf$ is Mann–Whitney's statistic for the conditional distributions of $Y$ given $X_s$ and $X_f\cdot$.
For $k=l$, $n_{s\cdot}=n_{\cdot t}=N/k$ for all $s$ and $t$, put $\widehat p_{st}=U_{st}/n_{s\cdot}n_{f\cdot}$; then
$$ \tau=2\Biggl[\frac{\sum_{s=1}^k\sum_{t=1}^{s-1}\widehat p_{st}}{1/2k(k-1)}-\frac12\Biggr]. $$
The first term in the brackets is the mean value of the normalized Mann–Whitney's statistic over all paired comparisons of conditional distributions of $Y$.