Select $n$ numbers from a set $\{1,2,...,U\}$, $y_i$ is the $i$th number selected, and $x_i$ is the rank of $y_i$ in the $n$ numbers. The rank is the order of the a number after the $n$ numbers are sorted in ascending order.
We can get $n$ data points $(x_1, y_1), (x_2, y_2), ..., (x_n, y_n)$, And a best fit line for these data points can be found by linear regression. $r_{xy}$ (correlation coefficient) is the goodness of the fit line, I want to calculate $E(r_{xy})$ or $E(r_{xy}^2)$ (correlation of determination).