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{\bf Zoran \v Suni\'k}
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{\bf Self-Describing Sequences and the Catalan Family Tree}
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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.

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