PS: Distinct Permutation-based Sums

[Leroy Quet]

>>....

I wrote in a reply to Max's post:


>
>Uggg, you are right.
>
>So, typing without thinking...
>
>Can we establish a sequence with undistiguishable (by the definition 
>implied here) m's?
>
>We could define, for this purpose, a(m), if the m-permutation is 
>undistinguishable, arbitrarily as the lowest positive(/nonnegative) 
>integer not among a(1),a(2),...,a(m-1).
>
>So, the undistinguished m's would form the sequence:
>
>1, 4,...
>
>Is this sequence infinite?
>(For all I know, EVERY m >= 4 is undistinguished with the a's as 
>redefined.)
>
>
>And the a-sequence then would progress as (with 0 = a(1))
>
>0, 1, 3, 2,...
>
>Ideally, the a-sequence would then form an interesting permutation of the 
>nonnegative integers. (But then we would need there to be an infinite 
>number of indistinguished m's, given how the a-sequence is now defined, I 
>believe.)



Of course, I must point out that I realize that the indistinguisable 
sequence being infinite does not necessarily imply that the a-sequence is 
a permutation of the nonnegative integers.
Likely not actually, perhaps. Because the a-sequence would probably rise 
very fast, skipping lots of integers.
But then there would be probably a lot of indistinguishable m's {if not 
all m's >= 4}...


>thanks,
>Leroy
>  Quet