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:

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