Combinatorial Constructions for Sifting Primes and Enumerating the Rationals

TL;DR

This paper introduces combinatorial bijections between rational numbers and rooted trees, providing new algorithms for prime sifting and generating reduced rationals, advancing enumeration and structural understanding of rationals.

Contribution

It presents two novel combinatorial bijections linking rationals and rooted trees, leading to new algorithms for prime sifting and rational enumeration.

Findings

New bijective mappings between integers, rationals, and rooted trees.

A novel algorithm for prime sifting based on combinatorial structures.

Refinements of the Wilf-Calkin algorithm for generating reduced rationals.

Abstract

We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that such mappings achieve much more than enumeration of rooted trees. We discuss two related structural bijections. The first corresponds to a bijective map between integers and rooted trees. The first bijection also suggests a new algorithm for sifting primes. The second bijection extends the first one in order to map rational numbers to a family of rooted trees. The second bijection suggests a new combinatorial construction for generating reduced rational numbers, thereby producing refinements of the output of the Wilf-Calkin[1] Algorithm.