  This paper establishes connections between EW-tableaux and parallelogram
polyominoes by using recent research regarding the sandpile model on the
complete bipartite graph. This paper presents and proves a direct bijection
between rectangular EW-tableaux and labelled ribbon parallelogram polyominoes.
The significance of this is that allows one to move between these objects
without the need for `recurrent configurations', the central object which
previously tied this work together. It introduces the notion of a marked
rectangular EW-tableaux that exactly encode all recurrent configurations of the
sandpile model on the complete bipartite graph. This shows how
non-cornersupport entries that featured in previous work can be utilized in a
simple but important way in relation to EW-tableaux. It lifts the bijection
between rectangular EW-tableaux and labelled ribbon parallelogram polyominoes
to a bijection between marked rectangular EW-tableaux and labelled
parallelogram polyominoes. This bijection helps us to fully understand the
aspects of these very different objects that are, in a sense, different sides
of the same coin.
