On the number of monotone sequences

One of the most classical results in Ramsey theory is the theorem of Erd\H{o}s and Szekeres from 1935, which says that every sequence of more than $k^2$ numbers contains a monotone subsequence of length $k+1$. We address the following natural question motivated by this result: Given integers $k$ and $n$ with $n\ge k^2+1$, how many monotone subsequences of length $k+1$ must every sequence of $n$ numbers contain? We answer this question precisely for all sufficiently large $k$ and $n\le k^2+c{k}^{3\mathrm{/}2}\mathrm{/}\log k$, where $c$ is some absolute positive constant.