Permutation Distances

Probability Level 5

xi x_i is a permutation of {1,2,,402} \{ 1, 2, \ldots, 402 \} . What is the largest value of S=x1x2x3x4x402S = | \ldots | | |x_1 - x_2| -x_3| - x_4 | - \ldots -x_{402} | ?

Details and assumptions

| \cdot | is the absolute value function, which satisfies

x={xx0xx<0 |x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases} \\

For example, 5=5,7=7 | 5 | = 5, |-7| = 7 .

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