Problem

Source: Iran 3rd round 2009 - final exam problem 2

Tags: combinatorics unsolved, combinatorics



Permutation $\pi$ of $\{1,\dots,n\}$ is called stable if the set $\{\pi (k)-k|k=1,\dots,n\}$ is consisted of exactly two different elements. Prove that the number of stable permutation of $\{1,\dots,n\}$ equals to $\sigma (n)-\tau (n)$ in which $\sigma (n)$ is the sum of positive divisors of $n$ and $\tau (n)$ is the number of positive divisors of $n$. Time allowed for this problem was 75 minutes.