# 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

```