Given a permutation s of {1,..,n}, define p to be the least such that s(i) = i for i > p. Suppose s is not the identity; then p > 0. Define q,r by q = s(p), s(r) = p; then q,r < p. Define a distinct permutation t by t(i) = s(i) for i <> p,r; t(p) = p, t(r) = q; Then f(t) - f(s) = c_p c_p + c_q c_r - c_p c_q - c_p c_r = (c_p - c_q)(c_p - c_r) > 0. Therefore f(s) is not a maximum. WFL On 2/6/07, Daniel Asimov <dasimov@earthlink.net> wrote:
Given fixed real numbers c_1 < c_2 < . . . < c_n, define
f(sigma) = Sum_{1 <= j <= n} c_j c_sigma(j)
for any permutation sigma of {1,...,n}.
Prove that sigma = id_{1,...n} is the unique permutation that maximizes f(sigma).
--Dan
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