The Genesis Sequence, Tree Records and Endofunctions

Enrica Duchi
(Paris, France) 		Adrián Lillo
(Sevilla, Spain) 		Pablo Puerto
(Sevilla, Spain) 		Mercedes Rosas
(Sevilla, Spain) 		Stefan Trandafir
(Vancouver, Canada)

We present bijections connecting tree records, the girth of a connected endofunction, and the genesis sequence (the first sequence in OEIS). Using these, we derive generating functions for tree and forest record numbers in terms of Cayley's tree function and give a new proof of Cayley's forest formula.

In Srečko Brlek and Luca Ferrari: Proceedings of the 14th edition of the conference on Random Generation of Combinatorial Structures (GASCom 2026), Malosco (BZ), Italy, 8-12 June 2026, Electronic Proceedings in Theoretical Computer Science 445, pp. 87–94.
This is an extended abstract of https://arxiv.org/abs/2601.07938. This work is dedicated to Neil J. A. Sloane, renowned number-crusher, for his creation and tireless curation of the Online Encyclopedia of Integer Sequences.
Published: 7th June 2026.
ArXived at: https://dx.doi.org/10.4204/EPTCS.445.11 bibtex PDF
References in reconstructed bibtex, XML and HTML format (approximated).
Comments and questions to: eptcs@eptcs.org
For website issues: webmaster@eptcs.org