Growth rates of permutations with given descent or peak set

Given a set $I\subseteq N$, consider the sequences $\left\{d_n\left(I\right)\right\},\left\{p_n\left(I\right)\right\}$ where for any $n$, $d_n\left(I\right)$ and $p_n\left(I\right)$ respectively count the number of permutations in the symmetric group $S_n$ whose descent set (respectively peak set) is $I\cap [n-1]$. We investigate the growth rates $gr{d}_n\left(I\right)={lim}_{n\to \infty }{\left(d_n\left(I\right)/n!\right)}^{1/n}$ and $gr{p}_n\left(I\right)={lim}_{n\to \infty }{\left(p_n\left(I\right)/n!\right)}^{1/n}$ over all $I\subseteq N$. Our main contributions are two-fold. Firstly, we prove that the numbers $gr{d}_n\left(I\right)$ over all $I\subseteq N$ are exactly the interval $\left(0,2/\pi \right)$. To do so, we construct an algorithm that explicitly builds $I$ for any desired limit $L$ in the interval. Secondly, we prove that the numbers $gr{p}_n\left(I\right)$ for periodic sets $I\subseteq N$ form a dense set in $\left(0,1/\sqrt[3]{3}\right)$. We do this by explicitly finding, for any prescribed $L$ in the interval, a set $I$ whose corresponding growth rate is arbitrarily close to $L$.