Kevin Ryde: That's a big table of permutations!  Oh, "AYT" is what I've been calling A020650/A020651.  They're in the Index for David W. Wilson creator of the sequences.  I'll double-check and add reference to Shen Yu-Ting, and the later D.N. Andreev.

Your A245325 etc are good bit flips, so right-to-left readings, as you note I think.  It's only neglect on my part not to mention such options somewhere in my write-up (such as its present state is etc etc).

(You were/are already at reviewed state here, so don't let me put you off calling a halt and mustering relationships later ...)

Yosu Yurramendi: That's right ('Lexicographically smallest permutation which maps etc etc...'). which(A162911 == 3) [1]   5   7   8  12  22  30  34  50  90 122 138 202. which(A162912 == 3) [1]   4   6  11  15  17  25  45  61  69 101 181 245
So, a(5)=4, a(7)=6, a(8)=11, ... But it is difficult for me to formalize such a definition. This situation is similar to A153151 (Calkin-Wilf), A153141 (Stern-Brocot system), A154448 (Bird system) and their inverses.

Kevin Ryde: Ah yes.  The easiest lex-smallest A065190 "AYT" is swap pairs, where you've been before :).  CW A153151 notes it is lex-smallest too.  Then Drib here would be the remaining "low-to-high tree" ... if true.

I didn't know SB A153141 or Bird A154448.  Are you here bit-reversal of the Bird one, A059893 style?  So a(n) = A059893(A154448(A059893(n))) ?

Yosu Yurramendi: Thank you very much for your observations and proposals.
*** This text editor is quite rough to show you in a ordered and comprehensible way the permutations between numerator and denominator of the well-known enumeration systems of positive rationals. I propose you to visit at http://oeis.org/wiki/User:Yosu_Yurramendi#Permutations_between_numerator_and_denominator_of_systems (I feel a bit embarrassed because the names of new two systems; they are due to A. Karttunen http://oeis.org/wiki/Index_to_OEIS:_Section_Fo#fraction_trees . Morever, they are not considered in your work https://download.tuxfamily.org/user42/rationals/rationals.pdf).
*** Permutation A059893(A154448(A059893(n))) is not at OEIS. I didn't find any particular property for this permutation. I think your question is about a permutation Axxxxxx such that A334998 = Axxxxxx(A154448(Axxxxxx(n))).  It's difficult for me to solve this equation. In the same way: A334999 = Ayyyyyy(A154447(Ayyyyyy(n))). Maybe AXXXXXX and AYYYYYY are inverse to each other: A000027 = A334998(A334999) = Axxxxxx(A154448(Axxxxxx(Ayyyyyy(A154447(Ayyyyyy(n)))))).
*** Sorry, I don't know what 'AYT' means.

Kevin Ryde: Let me hope I have this the right way around:
A162911(5)=3, so a(5) is to be a location in A162912 where value 3.  The first such is at A162912(4)=3.  So set a(5)=4.
A162911(7)=3, so a(7) another location of a 3.  Already used A162912(4)=3, but the next one is A162912(6)=3.  So set a(7)=6.
And so on, whenever want a 3 (or whatever value), take the next one (smallest position).

The computer says this holds throughout your b-files.  Assuming it's true, there should be enough pattern in the Drib descents to prove it.  But leave it for future work if it won't spring immediately from your construction :-).

Incidentally, there'll be a low-bits bit-flip rule lurking.  0<->1 flip of a low bit pattern ...1010 in n.  I think A087230 gives the length of flipping, except must reduce it so never touch or go above the highest 1-bit (which defines the "row").  Yet more work eh ...

Yosu Yurramendi: Thanks, Kevin. I don't understand what you mean with ' The data values look like the earliest (not yet used) position of each target value'. Can you give me an example?