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#### Olinguito

##### Well-known member

- Apr 22, 2018

- 251

1. For each $\pi\in S_n$,

$$\sum_{i=1}^n\,(\pi(i)-i)\ =\ 0.$$

2. If

$$\sigma_\pi\ =\ \sum_{i=1}^n\,\left|\pi(i)-i\right|$$

for each $\pi\in S_n$, then $\sigma_\pi$ is an even number.

Bonus challenge: Find $\displaystyle\max_{\pi\in S_n}\,\sigma_\pi$.