On the existence of permutations conditioned by certain rational functions

TL;DR

This paper proves conjectures by Z.-W. Sun regarding the existence of permutations constrained by specific rational functions and characterizes all integer values of an inverse difference rational function.

Contribution

It provides a complete proof of Sun's conjectures and characterizes the integer values of a particular rational function, combining theoretical analysis and computational methods.

Findings

Confirmed several of Sun's conjectures on permutation existence.

Characterized all integer values of the inverse difference rational function.

Used both mathematical analysis and computer-assisted search.

Abstract

We prove several conjectures made by Z.-W. Sun on the existence of permutations conditioned by certain rational functions. Furthermore, we fully characterize all integer values of the "inverse difference" rational function. Our proofs consist of both investigation of the mathematical properties of the rational functions and brute-force attack by computer for finding special permutations.