  We consider the number of passes a permutation needs to take through a stack
if we only pop the appropriate output values and start over with the remaining
entries in their original order. We define a permutation $\pi$ to be $k$-pass
sortable if $\pi$ is sortable using $k$ passes through the stack. Permutations
that are $1$-pass sortable are simply the stack sortable permutations as
defined by Knuth. We define the permutation class of $2$-pass sortable
permutations in terms of their basis. We also show all $k$-pass sortable
classes have finite bases by giving bounds on the length of a basis element of
the permutation class for any positive integer $k$. Finally, we define the
notion of tier of a permutation $\pi$ to be the minimum number of passes after
the first pass required to sort $\pi$. We then give a bijection between the
class of permutations of tier $t$ and a collection of integer sequences studied
by Parker. This gives an exact enumeration of tier $t$ permutations of a given
length and thus an exact enumeration for the class of $(t+1)$-pass sortable
permutations. Finally, we give a new derivation for the generating function in
Parker's thesis and an explicit formula for the coefficients.
