In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), the map $\omega \mapsto \sigma(\omega)$ is bijective from $\Omega_{n}$ to $S_{n}$ and induces a bijection between the event $(X_{n} = m)$ and the set of permutations of $S_{n}$ having $m-1$ ascents. We denote by $A_{n,m}$ the number of elements of $S_{n}$ with $m$ ascents. Let $m \in \llbracket 1, n \rrbracket$. Determine, for any integer $n \geqslant 2$ and any $m \in \llbracket 0, n-1 \rrbracket$, the number $A_{n,m}$ of permutations of $S_{n}$ having $m$ ascents.
In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), the map $\omega \mapsto \sigma(\omega)$ is bijective from $\Omega_{n}$ to $S_{n}$ and induces a bijection between the event $(X_{n} = m)$ and the set of permutations of $S_{n}$ having $m-1$ ascents. We denote by $A_{n,m}$ the number of elements of $S_{n}$ with $m$ ascents.
Let $m \in \llbracket 1, n \rrbracket$. Determine, for any integer $n \geqslant 2$ and any $m \in \llbracket 0, n-1 \rrbracket$, the number $A_{n,m}$ of permutations of $S_{n}$ having $m$ ascents.