Summary

The four different orderings of the  A181897(5,6) = 24 permutations (compare this) of {1,2,3,4,5} with a complete 5-cycle:
Lexicographic, reverse lexicographic, colexicographic and reverse colexicographic order.
Reflected orderings are not possible in this case, because the reflected permutations are not among the 24 permutations that shall be ordered.

The inversion vectors are shown in red to the left of the permutations.
The index numbers of the permutations (compare  A195663) are shown in light gray to the left of the vectors.

Only the reverse colexicographic order allows an infinite enumeration of all finite permutations.

Lexicographic and reverse lexicographic order don't correspond to any order of the inversion vectors.

This file on OEIS-Wiki: File:Orderings; permutations 5-cycle.svg

Compare:

Published by Watchduck under the following licences:

I, the copyright holder of this work, hereby publish it under the following licenses:

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

This file is licensed under the Creative Commons Attribution 3.0 Unported license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.

https://creativecommons.org/licenses/by/3.0CC BY 3.0 Creative Commons Attribution 3.0 truetrue

You may select the license of your choice.