$\pi\in S_n$, $\mathrm{desc}(\pi):=\{i\in[n-1]: \pi_{i+1}<\pi_i\}$. $S\subset[n-1]$ and $S$ of size $k$. Could you help me to calculate the following numbers, please?
1) The number of permutations $\pi\in S_n$ with $\mathrm{desc}(\pi)\supseteq S$.
2) The number of permutations $\pi\in S_n$ with $\mathrm{desc}(\pi)=S$.