  We describe a new method for finding patterns in permutations that produce a
given pattern after the permutation has been passed once through a stack. We
use this method to describe West-3-stack-sortable permutations, that is,
permutations that are sorted by three passes through a stack. We also show how
the method can be applied to the bubble-sort operator. The method requires the
use of mesh patterns introduced by Br\"and\'en and Claesson (2011), as well as
a new type of generalized pattern we call a decorated pattern.
