IMC 2016 Problem 5

Let $S_n$ denote the set of permutations of the sequence $(1,2,\dots, n)$ . For every permutation $\pi=(\pi_1, \dots, \pi_n)\in S_n$ , let $\mathrm{inv}(\pi)$ be the number of pairs $1\le i < j \le n$ with $\pi_i>\pi_j$ ; i. e. the number of inversions in $\pi$ . Denote by $f(n)$ the number of permutations $\pi\in S_n$ for which $\mathrm{inv}(\pi)$ is divisible by $n+1$ . Prove that there exist infinitely many primes $p$ such that $f(p-1)>\frac{(p-1)!}{p}$ , and infinitely many primes $p$ such that $f(p-1)<\frac{(p-1)!}{p}$ .

(Proposed by Fedor Petrov, St. Petersburg State University)