Enumerating linear systems on graphs

To play the dollar game on a graph, start by assigning to each vertex a number of dollars of either wealth or debt.  From this initial state, called a divisor, the vertices lend and borrow with their neighbors according to chip-firing rules in an attempt to reach a debt-free state. The set of all possible winning states is the complete linear system of the divisor.  We are interested in determining its cardinality.  In the figure below, each vertex represents one of the 201 winning positions resultingfrom giving ten dollars to one vertex of the cycle graph on five vertices.  This is joint work with Sarah Brauner and Forrest Glebe.