For $1 \le i \lt n$, let $m_i$ be the number of inversions $(i,j), i \lt j \le n,$ in permutation $ \sigma$. Let $ \sigma_i = (i+m_i,i+m_i-1)..(i+1,i)$.
How to prove that $\sigma=\sigma_1...\sigma_{n-1}$?
$\sigma$ is a n-permutation.
Inversions of a permutation are the pairs $(i,j)$ such that $1 \le i \lt j \le n$ and $ \sigma (i) \gt \sigma (j)$