The genesis sequence, tree records and endofunctions

TL;DR

The paper builds a network of bijections linking tree-records, endofunction girth, and the genesis sequence to reveal deep combinatorial structure. By translating tree-records into endofunctions and vice versa, it derives closed-form tree-record counts, generates exact generating functions via the Cayley tree function, and furnishes a new proof of Cayley’s forest formula. The work also connects the genesis sequence to non-root tree records through catalysts, and ties forest weights to symmetric-group statistics, enriching the combinatorial landscape around labeled trees and endofunctions. Overall, it unifies several classical objects (Cayley trees, forest formula, genesis sequence) under a cohesive bijective framework with explicit algebraic tools. The results have potential implications for enumerative combinatorics, algebraic combinatorics, and the study of integer-sequence structures in the OEIS.

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.

Figures (9)

• Figure 1: The bijection of Theorem \ref{['thm:connected endofunctions']}. The distinguished record $v = 9$ is circled. The path from $v$ to the root (on the left) and the only cycle in the resulting endofunction (on the right) are highlighted.
• Figure 2: The repeated application of the bijection in Theorem \ref{['Thm:endo_girth']}. The records are highlighted. Note that the leftmost endofunction has at least $4$ records and girth $1$, and the rightmost endofunction has at least $1$ record and girth $4$.
• Figure 3: The bijection of Theorem \ref{['thm:catalysts']}.
• Figure 4: On the left, two rooted trees with a distinguished vertex each. On the right, the corresponding catalyst under the bijection of Lemma \ref{['lem:Riordan_Sloane']}.
• Figure 5: The bijection $\phi$. In this example, $v = 8$, $l = 1$ and $(x_0, x_1, x_2, x_3) = (g, x_1, v, p) = (1,6, 8, 4)$.

Theorems & Definitions (56)

• Theorem 1.1
• proof
• Corollary 1.2
• Theorem 1.3
• proof
• Corollary 1.4
• proof
• Corollary 1.5
• proof
• Corollary 1.6