Special property of corresponding sequences

• Mar 27th 2012, 03:48 AM
princeps
Special property of corresponding sequences
For any sequence s consisting of 1's and 2's, let r(s) denote the length of the nth run of same symbols in s .

There is a unique nontrivial sequence s such that $s(1) = 1$ and $r(r(s(n))) = s(n)$ for all $n$ :

$s = (1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, . . .)$

$r(s) = (2, 1, 2, 2, 1, 2, 1, 1, 2 , 2 ,1, 2, 2, 1, 1, 2, 1, . . .)$

Question : Prove or disprove that every segment of r(s) is a segment of s.

For example , the initial segment 1121 of s occurs in r(s) beginning at the 14th term.

P.S.

I am interested in hints (not full solution)
• Mar 27th 2012, 05:40 AM
emakarov
Re: Special property of corresponding sequences
Quote:

Originally Posted by princeps
Question : Prove or disprove that every segment of r(s) is a segment of s.

For example , the initial segment 1121 of s occurs in r(s) beginning at the 14th term.

The question seems to be about finding segments of r(s) in s, not the other way around.

The sequence r(s) does not need to have the property that the maximum possible length of the nth run is 2.
• Mar 27th 2012, 06:27 AM
princeps
Re: Special property of corresponding sequences
Have you any idea how to approach to this problem ?
• Mar 27th 2012, 06:35 AM
emakarov
Re: Special property of corresponding sequences
Quote:

Originally Posted by princeps
Have you any idea how to approach to this problem ?

Yes.
Quote:

Originally Posted by emakarov
The sequence r(s) does not need to have the property that the maximum possible length of the nth run is 2.