We denote by $S_{n}$ the set of permutations of $\llbracket 1, n \rrbracket$. We represent an element $\sigma$ of $S_{n}$ by the finite sequence $(\sigma(1), \sigma(2), \ldots, \sigma(n))$ and we call an ascent (respectively descent) of $\sigma$ an ascent of this sequence. For any integer $m$, we denote by $A_{n,m}$ the number of elements of $S_{n}$ with $m$ ascents.
Let $n \geqslant 2$. Determine $A_{n,0}$, $A_{n,n-1}$ and $A_{n,m}$ for $m \geqslant n$.