Counting and naming connection islands on a grid of conductors

Abstract

Consider a grid of T horizontal and T vertical information conductors. At each crossing point it is possible to either connect a horizontal to a vertical conductor or just let the connection be open. When connected at a crossing point, both conductors share the same information state through-their lengths. When several conductors are mutually connected at their crossing points, an island of connections is formed. A requirement for an island is that there be representative conductors of both kinds. All conductors associated with an island share the same information state. For the case of several islands, each island's conductors are mutually exclusive of any other island's conductors. In fact, an island can be uniquely named by its conductors. As a further requirement on the conductors, the grid is assumed to be completely utilized with no idle conductors. It is under the above conditions that we develop the general combinatorial equations for counting the number of ways S sets of islands can exist on a T × T grid of conductors. The restraints of the problem introduce interesting combinatorial aspects. We also discuss methods of counting connections and naming sets of islands.