The Genesis Sequence, Tree Records and Endofunctions

2026-06-12

AI Summary

This work establishes, for the first time, a bijection between the OEIS “Genesis sequence,” the number of records in rooted trees, and the girth of connected endofunctions. By constructing this combinatorial bijection and integrating it with Cayley tree functions and generating function techniques, the authors derive exact generating functions for the distribution of records in both trees and forests. Building upon this unified framework, they provide a concise and novel proof of the classical enumeration formula for Cayley forests. The approach offers fresh insight and a coherent interpretation of these fundamental combinatorial structures, revealing deeper connections among seemingly disparate objects in enumerative combinatorics.

Abstract

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.

Problem

• tree records
• endofunctions
• girth
• genesis sequence
• generating functions

Innovation

• bijections
• tree records
• endofunctions
• generating functions
• Cayley's forest formula