The question:What permutation(s)^{†}give the maximum values for ?

Definition:Let be a map such that^{‡}

Explanation:I want to maximize (when given a particular ). If you think if as distinct elements in a sequence (of length ), and the permutation as moving them to new locations (i.e. shuffling the sequence), then would represents a sort of metric for how "different" the new permuted sequence would be from the original (in order) sequence.

For small I though about making a computer program to brute-force the problem; (i.e. compute for every ). However, since and I'm really interested in larger e.g. , that method becomes infeasible. If there are permutations I could rule out -- e.g. for (i.e. the identity map), , a minimum, not a maximum -- maybe a computer program could analyze the remaining permutations. I also though about just randomly computing for different permutations and going with a high-water mark algorithm on those.

Any suggestions on how to approach this problem, or where I could study/look would be appreciated.

^{†}

See Permutation - Wikipedia, the free encyclopedia

^{‡}

I used bracket notation instead because of the nested absolute values.