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A problem in counting permutations

Source: Problem 1, Brazilian Olympic Revenge 2005

June 1, 2005
combinatorics unsolvedcombinatorics

Problem Statement

Let S={1,2,3,…,n}S=\{1,2,3,\ldots,n\}, nn an odd number. Find the parity of number of permutations σ:S⇒S\sigma : S \Rightarrow S such that the sequence defined by a(i)=∣σ(i)−i∣a(i)=|\sigma(i)-i| is monotonous.
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