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Permutation Minimum and Maximum

Source: China TST 1999, problem 6

May 22, 2005
combinatorics unsolvedcombinatorics

Problem Statement

For every permutation τ \tau of 1,2,…,10 1, 2, \ldots, 10, \tau \equal{} (x_1, x_2, \ldots, x_{10}), define S(\tau) \equal{} \sum_{k \equal{} 1}^{10} |2x_k \minus{} 3x_{k \minus{} 1}|. Let x_{11} \equal{} x_1. Find I. The maximum and minimum values of S(τ) S(\tau). II. The number of τ \tau which lets S(τ) S(\tau) attain its maximum. III. The number of τ \tau which lets S(τ) S(\tau) attain its minimum.
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