  In an exercise in the first volume of his famous series of books, Knuth
considered sorting permutations by passing them through a stack. Many
variations of this exercise have since been considered, including allowing
multiple passes through the stack and using different data structures. We are
concerned with a variation using pop-stacks that was introduced by Avis and
Newborn in 1981. Let $P_k(x)$ be the generating function for the permutations
sortable by $k$ passes through a pop-stack. The generating function $P_2(x)$
was recently given by Pudwell and Smith (the case $k=1$ being trivial). We show
that $P_k(x)$ is rational for any $k$. Moreover, we give an algorithm to derive
$P_k(x)$, and using it we determine the generating functions $P_k(x)$ for
$k\leq 6$.
