  Rothe diagrams are diagrams which track inversions of a permutation. We
define six main properties that Rothe diagrams fulfill: the southwest, dot,
popping, numbering, step-out avoiding, and empty cell gap rules. We prove that
-- given an arbitrary bubble diagram -- four different subsets of these
properties provide sufficient criteria for the diagram to be a Rothe diagram.
We also prove that when a set of ordered, freely floating, non-empty columns
satisfy the numbering and step-out avoiding rules, then they can be arranged
into a Rothe diagram.
