This rather simple result is a humble acknowledgement of the great work in finite-size random matrix theory (RMT) by Prof. Gernot Akemann and team at Uni Bielefeld. For finding the ordered densities, a straightforward recursive formulation, in terms of the MeijerG function, based on the work of Alberto Zanella at CNR in Italy, is utilized.
Note: Several integration formulas for the MeijerG function are known, e.g., see the MeijerG function reference.Although the expressions are complex, they can be numerically evaluated quite easily via Mathematica or MATLAB. It amazes me that these finite-size RMT densities are even analytically approachable, although, undoubtedly, the asymptotic RMT theory is "more elegant."
The theorem is as follows. See below for Mathematica code and numerical simulations.
Theorem
Second projection notation: Let set $s := \{(1,2), (3,4), (5,6)\}$ be a set containing three tuples. Then, $\pi_2(s)$ is the second projection of $s$ given by $\pi_2(s) = \{2, 4, 6\}.$