Cycles and divergent trajectories for a class of permutation sequences

TL;DR

This paper studies the behavior of permutation sequences generated by functions on natural numbers, providing bounds for when these sequences form cycles or diverge, with theoretical and computational insights.

Contribution

It introduces bounds for cycles and divergent trajectories in a specific class of permutations, combining theoretical analysis with computational methods.

Findings

Derived bounds for cycle lengths in permutation sequences.

Established criteria for divergence of permutation trajectories.

Provided computational tools to analyze permutation behaviors.

Abstract

Let $f$ be a permutation from $\mathbb{N}_0$ onto $\mathbb{N}_0$. Let $x\in\mathbb{N}_0$ and consider a (finite or infinite) sequence $s= (x,f(x),f^2(x),\cdots)$. We call $s$ a permutation sequence. Let $D$ be the set of elements of $s$. If $D$ is a finite set then the sequence $s$ is a cycle, and if $D$ is an infinite set the sequence $s$ is a divergent trajectory. We derive theoretical and computational bounds for cycles and divergent trajectories for a defined class of permutations.