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Kosovo Mathematical Olympiad, #5. (Grade 12) [Permutations]

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March 13, 2011
combinatorics proposedcombinatorics

Problem Statement

Let n>1n>1 be an integer and SnS_n the set of all permutations π:{1,2,⋯ ,n}→{1,2,⋯ ,n}\pi : \{1,2,\cdots,n \} \to \{1,2,\cdots,n \} where π\pi is bijective function. For every permutation π∈Sn\pi \in S_n we define:
F(π)=∑k=1n∣k−π(k)∣  and  Mn=1n!∑π∈SnF(π) F(\pi)= \sum_{k=1}^n |k-\pi(k)| \ \ \text{and} \ \ M_{n}=\frac{1}{n!}\sum_{\pi \in S_n} F(\pi) where MnM_n is taken with all permutations π∈Sn\pi \in S_n. Calculate the sum MnM_n.
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