# 2020-09 Displacement of permutations

For a permutation $$\pi: [n]\rightarrow [n]$$, we define the displacement of $$\pi$$ to be $$\sum_{i\in [n]} |i-\pi(i)|$$.
For given $$k$$, prove that the number of even permutations of $$[n]$$ with displacement $$2k$$ minus the number of odd permutations of $$[n]$$ with displacement $$2k$$ is $$(-1)^{k}\binom{n-1}{k}$$.

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# Notice on POW 2020-08

Due to a technical issue, POW 2020-08 was posted on Monday. Correspondingly, the due date for POW 2020-08 is postponed to May 29, Sat. (by noon).

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# 2020-08 Geometric action revisited

In the problem 2019-08 (https://mathsci.kaist.ac.kr/pow/2019/2019-08-group-action/), we considered a group G acting by isometries on a proper geodesic metric space X properly discontinuously and cocompactly. Such an action is called a geometric action. The conclusion was that a geometric action leads to that G is finitely generated.

Would this conclusion still hold in the case the space X is not necessarily proper?

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# Solution: 2020-07 Perfect square

Suppose that $$x, y, z$$ are positive integers satisfying
$0 \leq x^2 + y^2 – xyz \leq z+1.$
Prove that $$x^2 + y^2 – xyz$$ is a perfect square.

The best solution was submitted by 임상호 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-07.

Another solution was submitted by 김기수 (수리과학과 2018학번, +3), 홍의천 (수리과학과 2017학번, +3)

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# 2020-07 Perfect square

Suppose that $$x, y, z$$ are positive integers satisfying
$0 \leq x^2 + y^2 – xyz \leq z+1.$
Prove that $$x^2 + y^2 – xyz$$ is a perfect square.

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