Is there a formula for the number of permutations of $[n]$ with $k$ peaks?
Here is some information on the OEIS about these numbers, but no such formula is given. I'm sure such a formula exists due to the following question posed in Miklós Bóna's book, Combinatorics of Permutations (p.40, Q.33). Unfortunately the Google book is only a preview and the required solutions are missing.
EDIT: Just to clarify, I understand that an element $a_k$ of a permutation $[n]=123\cdots n$ is called a peak when $a_{k-1}<a_k$ and $a_k>a_{k+1}$. We define $a_0=a_{n+1}=0$.
You can get a recurrence relation for these numbers by defining the numbers $a_{nk}^{\pm\pm}$, where the first sign is $+$ for permutations beginning with a peak and $-$ otherwise, and the second sign likewise for ending with a peak. Thus, for instance, $a_{nk}^{++}$ counts the number of permutations of $[n]$ with $k$ peaks that begin and end with a peak. (Clearly $a_{nk}^{+-}=a_{nk}^{-+}$.) Then by considering the possible places in which you can insert $n+1$ in a permutation of $[n]$ you get
$$ a_{n+1,k}^{++}=\sum_{i=0}^n\left(\sum_{j=0}^{k-1}a_{ij}^{+-}a_{n-i,k-1-j}^{-+}+2\sum_{j=0}^ka_{ij}^{++}a_{n-i,k-j}^{-+}+\sum_{j=0}^{k+1}a_{ij}^{++}a_{n-i,k+1-j}^{++}\right) $$
and similar relations for $a_{n+1,k}^{-+}$ and $a_{n+1,k}^{--}$, with suitable initial and boundary conditions.