Distinct Permutation-based Sums

[Leroy Quet]

>Leroy Quet wrote:
>
>>Consider the permutations of {1,2,3,...,n}.
>>Let [b(k)] be any one such permutation.
>>
>>
>>If we have an integer sequence {a(k)}, we can take the sum, for the 
>>particular permutation b,
>>
>>
>> sum{k=1 to n}  a(k)*b(k).
>>
>>  
>>
>If all such sums are distinct for different permutations, we say that 
>sequence {a(k)} distinguishes the permutations of order n.
>
>I know a programming contest problem asking to find such sequence for 
>n=5 with minimum value of
>A=sum{k=1 to n} a(k).
>And the answer is { 0, 2, 5, 21, 91 } with A=119.
>
>Similar sequence for n=6 is likely to be { 0, 1, 6, 32, 172, 950 } with 
>A=1161 (but I haven't checked it up).
>
>Another variation is to minimize
>max { a(k) }
>For n=5 it gives sequence { 0, 1, 8, 38, 87 }.


Interesting. But to clarify, the sequence I am asking about is fixed
for all values up to a(m-1), *then* a(m) is determined.

So, the first 4, say, terms would be the same for all permutations,
no matter how many terms (>= 4) are in them.


>
>>But, as I am guessing, 
>>
>>there is a specific infinite positive integer sequence {a(k)}, where
>>
>>a(1) = 1, 
>>
>>  
>>
>a(1)=0 looks more natural for me.

Possibly. If we add any same constant integer to  every term, the 
permutations are still distinguished for any particular m. (So the 1-to-0 
shift for a(1) could be a job for Superseeker.)
:)


Leroy


>
>>and for all m's >= 2,
>>
>>a(m) is the MINIMUM integer > a(m-1) such that:
>>
>>for each permutation of {1,2,3,...,m},
>>
>>   sum{k=1 to m}  a(k)*b(k)
>>
>>is DISTINCT among those sums based on all m! permutations of 1 to m, for 
>>every fixed m.
>>
>>  
>>
>Yeah, it deserves to be in OEIS.
>
>Regards,
>Max
>