  We address the following natural but hitherto unstudied question: what are
the possible linear extension numbers of an $n$-element poset? Let
$\mathbf{LE}(n)$ denote the set of all positive integers that arise as the
number of linear extensions of some $n$-element poset. We show that
$\mathbf{LE}(n)$ skews towards the "small" end of the interval $[1,n!]$. More
specifically, $\mathbf{LE}(n)$ contains all of the positive integers up to
$\exp\left(c\frac{n}{\log n}\right)$ for some absolute constant $c$, and
$|\mathbf{LE}(n) \cap ((n-1)!,n!]|<(n-3)!$. The proof of the former statement
involves some intermediate number-theoretic results about the Stern-Brocot tree
that are of independent interest.
