Iterating the Lehmer code on inversion sequences: Catalan fixed points and finite stabilization

We study an operator $Θ$ on finite integer sequences, where $Θ(σ)_i$ counts the entries to the left of $σ_i$ that are strictly smaller than $σ_i$. This operator is a variant of the so-called Lehmer code. For every sequence $σ$, the image $Θ(σ)$ is an inversion sequence, and the restriction of $Θ$ to