  We consider the avoidance of patterns in inversion sequences that relate
sorting via sorting machines including data structures such as pop stacks and
stacks. Such machines have been studied under a variety of additional
constraints and generalizations, some of which we apply here. We give the
classification of several classes of sortable inversion sequences in terms of
pattern avoidance. We are able to provide an exact enumeration of some of the
sortable classes in question using both classical approaches and a more recent
strategy utilizing generating trees.
