A Transform Of Seqs Involv Permutations

[Leroy Quet]

I might have written about this already a long while back. (It seems 
slightly familiar.) If so, I apologize.

Here is a transform which converts any sequence of positive integers and 
an infinite number of 1's into another sequence of positive integers and 
an infinite number of 1's.

(Perhaps some of you might want to submit new sequences generated this 
way.)



Start with a sequence,{a(k)}, of only positive integers and an infinite 
number of 1's.

Example:
1,1,2,1,2,3,1,2,3,4,1,...

Form the sequence {b(k)} (which is the permutation of the positive 
integers),
where b(k) = the a(k)th lowest positive integer not yet in the sequence b;
and b(1)=a(1).

1,2,4,3,6,8,5,9,11,13,7,...

Let {c(k)} be the inverse of {b(k)}.

1,2,4,3,7,5,11,6,8,...

Form the final sequence {d(k)}, where each d(k) is such that
c(k) = the d(k)th lowest positive integer not yet in the sequence c;
and d(1)=c(1).

1,1,2,1,3,1,5,1,1,...

So {1,1,2,1,2,3,1,2,3,4,...} is transformed into {1,1,2,1,3,1,5,1,1,...}.


Is there a more direct route to carrying out such a transform without 
finding the permutations? (I have not thought about this much.)

And this whole idea must not be new. What info is there about this, such 
as a preexisting name for the transform?


thanks,
Leroy Quet