Question 97
Let $S = \{1, 2, \dots, n\}$, $n \ge 2$, and let $f : S \to S$ be a bijective function distinct from the identity. Let $u = \sum_{k=1}^{n} |f(k) - k|$ and let $v$ be the number of ordered pairs $(a, b)$ of elements of $S$ such that $a \gt b$ and $f(a) \lt f(b)$. Show that $v \lt u \le 2v$, and that $u = 2v$ if and only if there do not exist positive integers $a \gt b \gt c$ such that $f(a) \lt f(b) \lt f(c)$.