Self-describing sequences and the Catalan family tree

We introduce a transformation of finite integer sequences, show that every sequence eventually stabilizes under this transformation and that the number of fixed points is counted by the Catalan numbers. The sequences that are fixed are precisely those that describe themselves – every term $t$ is equal to the number of previous terms that are smaller than $t$. In addition, we provide an easy way to enumerate all these self-describing sequences by organizing them in a Catalan tree with a specific labelling system.