The genesis sequence, tree records and endofunctions
Combinatorics
2026-01-14 v1
Abstract
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.
Keywords
Cite
@article{arxiv.2601.07938,
title = {The genesis sequence, tree records and endofunctions},
author = {Enrica Duchi and Adrián Lillo and Pablo Puerto and Mercedes Rosas and Stefan Trandafir},
journal= {arXiv preprint arXiv:2601.07938},
year = {2026}
}
Comments
26 pages, 9 figures