  We use a method for determining the number of preimages of any permutation
under the stack-sorting map in order to obtain recursive upper bounds for the
numbers $W_t(n)$ and $W_t(n,k)$ of $t$-stack sortable permutations of length
$n$ and $t$-stack sortable permutations of length $n$ with exactly $k$
descents. From these bounds, we are able to significantly improve the best
known upper bounds for $\displaystyle{\lim_{n\to\infty}\sqrt[n]{W_t(n)}}$ when
$t=3$ and $t=4$.
