  We prove a "decomposition lemma" that allows us to count preimages of certain
sets of permutations under West's stack-sorting map $s$. As a first
application, we give a new proof of Zeilberger's formula for the number of
2-stack-sortable permutations in $S_n$. Our proof generalizes, allowing us to
find an algebraic equation satisfied by the generating function that counts
2-stack-sortable permutations according to length, number of descents, and
number of peaks. The same method yields a recurrence relation for $W_3(n)$, the
number of 3-stack-sortable permutations in $S_n$. We compute $W_3(n)$ for $n\le
174$, extending the 13 terms of this sequence that were known before. We also
prove the first nontrivial lower bound for
$\lim\limits_{n\to\infty}W_3(n)^{1/n}$. Invoking a result of Kremer, we also
prove that $\lim\limits_{n\to\infty}W_t(n)^{1/n}\geq(\sqrt{t}+1)^2$ for all
$t\geq 1$, which we use to improve a result of Smith. Our computations allow us
to disprove a conjecture of B\'ona, although we do not yet know for sure which
one.
  We can refine our methods to obtain a recurrence for the number of
3-stack-sortable permutations in $S_n$ with $k$ descents and $p$ peaks. This
produces a large amount of evidence supporting a real-rootedness conjecture of
B\'ona. Using part of the theory of valid hook configurations, we give a new
proof of a $\gamma$-nonnegativity result of Br\"and\'en, which in turn implies
an older result of B\'ona. We then answer a question of the current author by
producing a set $A\subseteq S_{11}$ such that $\sum_{\sigma\in
s^{-1}(A)}x^{\text{des}(\sigma)}$ has nonreal roots. We interpret this as
partial evidence against the same real-rootedness conjecture of B\'ona that we
found evidence supporting. Examining the parities of the numbers $W_3(n)$, we
obtain strong evidence against yet another conjecture of B\'ona. We end with
some conjectures of our own.
