 Uniform linearized chord diagram

Pair $2n$ linearly ordered endpoints uniformly. Each pair is a chord; its larger endpoint ends a vertex block, with all endpoints after the previous right endpoint merged into that vertex. Orient the chord from its right-endpoint block to its left-endpoint block. Removing the last endpoint and its mate leaves a uniform smaller pairing. Reinserting the mate into one of $2n-1$ equally likely gaps chooses an existing block proportionally to its endpoint count, or the final gap creates a self-loop. This proves equivalence with the linearized chord diagram model.

An equivalent continuous representation draws $n$ independent pairs of independent uniform points on $[0,1]$, orders the pairs by their maxima $R_1<\cdots <R_n$, and calls their minima $L_i$. The $R_i^2$ are uniform order statistics; conditional on all the $R_i$, the $L_i$ are independent uniform points on $[0,R_i]$.