Distinct Permutation-based Sums

Leroy Quet qq-quet at mindspring.com
Sat Dec 13 03:15:01 CET 2003

>That's not true. a(m) may not exist. And it does not exist even for 
>small examples.
>Let a(1)=0, a(2)=1, a(3)=3. This sequence distinguishes all permutations 
>of order 3.
>But at the same time there is an equality
>a(3)*1 + a(1)*2 + a(4)*3 + a(2)*4 = a(2)*1 + a(3)*2 + a(4)*3 + a(1)*4
>that holds for ANY a(4).
>Hence,  a(4) can NOT be defined such that a(1), a(2), a(3),a(4) 
>distinguishes all permutations of order 4.

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 

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 


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