Erase to increment

[franktaw at netscape.net]

In fact, sequences with this property are closely related to fractal 
sequences, via what
I call the ordinal transform.  The ordinal transform of sequence a is 
the sequence b
where b(n) is the index of a(n) in the values of a that are equal to 
a(n).  E.g., if a
starts 1,1,2,1, b(4) will be 3, because a(4)  is the 3rd 1 in sequence 
a.

(As always with ordinals, there is the question of whether to start 
with 1 or 0.  I'm
assuming 1 here, but if the sequences you are dealing with include 0's, 
the 0-based
version is usually better.  They just differ by 1, in any event)

A sequence has the "erase to increment" property iff its ordinal 
transform is fractal.
(Recall that a sequence is fractal if removing the first instance of 
each value leaves
the original sequence.)  Applying the ordinal transform again will give 
the original
sequence.  (Generally, applying the ordinal transform three times is 
the same as
applying it once, so this is not a surprising result.)

Franklin T. Adams-Watters


-----Original Message-----
From: franktaw at netscape.net

I just submitted the following comment: 
 
... 
If all 1's are removed from the sequence, the resulting sequence b has 
b(n) = a(n)+1. 
...