The genesis sequence, tree records and endofunctions
Enrica Duchi, Adri\'an Lillo, Pablo Puerto, Mercedes Rosas, Stefan Trandafir
TL;DR
This paper uncovers deep combinatorial connections between tree records, endofunction girth, and the genesis sequence, deriving generating functions and offering a new proof for Cayley's forest formula.
Contribution
It introduces bijections linking tree records, endofunction girth, and the genesis sequence, and derives related generating functions and a novel proof of Cayley's forest formula.
Findings
Derived generating functions for tree and forest record numbers
Established bijections connecting key combinatorial concepts
Provided a new proof of Cayley's forest formula
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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Genome Rearrangement Algorithms · Algorithms and Data Compression