# A Transform Of Seqs Involv Permutations

Leroy Quet qq-quet at mindspring.com
Tue Dec 14 20:51:08 CET 2004

```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.)

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