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The genesis sequence, tree records and endofunctions

In this work, we present a series of bijections that reveal the deep connections between the concepts of tree records, the girth of a connected endofunction, and the genesis sequence, the first sequence in the OEIS. We use these results to derive the generating functions for the tree and forest record numbers, expressing them in terms of the Cayley's tree function. Finally, we provide a new proof for Cayley's forest formula.

preprint2026arXivOpen access
Enrica DuchiAdrián LilloPablo PuertoMercedes RosasStefan Trandafir
math.CO
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Enrica DuchiAdrián LilloPablo PuertoMercedes RosasStefan Trandafir

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