I think reasonable people could disagree as to what the "next" permutation is.

As seen by this survey of modules: Short survey of modules for combinations and permutations there is indeed disagreement. Lexicographic is the most common. Pari/GP's v2.6 changes include "permtonum/numtoperm now use the standard lexicographic numbering" so we might have some belief that that is a preferred ordering unless there are other requirements (e.g. Gray codes).

There are lots of possible reasonable orderings, e.g. Jorg Arndt's Algorithms book section 6.1 (Permutations) where numerous methods are shown, or Ruskey and William's Cool-lex ordering from 2009 -- noted earlier in Knuth's Volume 4 Fascicle 3 (2005).

I don't think most reasonable programmers would agree that "implementation defined and unlike every other implementation" is a good ordering. A number of modules on that list chose that so I guess there is disagreement. Some, like Math::Combinatorics, will give you a different ordering every run.