A permutation with a special property question

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SUMMARY

The discussion revolves around a mathematical problem involving permutations of the set {1, 2, ..., 2n}. Specifically, it addresses the condition that the absolute differences between consecutive elements in the permutation must be unique. The conclusion drawn is that if the even-indexed elements satisfy the condition 1 ≤ a_{2i} ≤ n for i = 1, 2, ..., n, then it must hold that a_{1} = a_{2n} + n. The participants express difficulty in proving this assertion, indicating a need for deeper exploration of the properties of permutations.

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Hells_Kitchen
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Hi there,
i was wondering if you had any thoughts on the following question:

Let [tex](a_{1}, a_{2}, ..., a_{2n})[/tex] be a permutation of [tex]{1, 2, ..., 2n}[/tex] so that [tex]|a_{i} - a_{i+1}| \neq |a_{j} - a_{j+1}|[/tex], whenever [tex]i \neq j[/tex].

Show that [tex]a_{1} = a_{2n} + n[/tex], if [tex]1 \leq a_{2i} \leq n[/tex] for [tex]i = 1,2, ..., n[/tex]
 
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still no ideas??
 
According to my calculations (proof) it's not possible.. Try proving this..
 
Last edited:

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