Distinct Permutation-based Sums
Leroy Quet
qq-quet at mindspring.com
Sat Dec 13 01:18:13 CET 2003
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).
But, as I am guessing,
there is a specific infinite positive integer sequence {a(k)}, where
a(1) = 1,
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.
I get a(2) = 2, and a(3) = 4.
For example, a(3) is not 3 because
1 * 1 + 2 * 3 + 3 * 2 =
1 * 2 + 2 * 1 + 3 * 3.
(the first integer in each product is of a(), the second integer is of
b().)
I am guessing that a(4) is at least 8, but I bet higher.
What is this sequence {a(k)}?
Does it exist in the OEIS as defined here or as something else seemingly
unrelated?
thanks,
Leroy Quet
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