Simple permutations with order $4n + 2$ by means of Pasting and Reversing

TL;DR

This paper explores the combinatorial structure and genealogy of simple permutations of order 4n+2 using Pasting and Reversing techniques, extending understanding of permutation dynamics beyond known cases.

Contribution

It introduces a novel combinatorial approach to analyze permutations of order 4n+2, expanding the genealogy framework with Pasting and Reversing methods.

Findings

Characterization of permutation structures of order 4n+2

Description of permutation genealogy using Pasting and Reversing

Insights into the dynamics of minimal 4n+2-orbits

Abstract

The problem of genealogy of permutations has been solved partially by Stefan (odd order) and Acosta-Hum\'anez \& Bernhardt (power of two). It is well known that Sharkovskii's theorem shows the relationship between the cardinal of the set of periodic points of a continuous map, but simple permutations will show the behaviour of those periodic points. Recently Abdulla et al studied the structure of minimal $4n+2$-orbits of the continuous endomorphisms on the real line. This paper studies some combinatorial dynamics structures of permutations of mixed order $4n+2$, describing its genealogy, using Pasting and Reversing.