Combinatorial Problems Suggested by the Statistical Mechanics of Domains and of Rubber-Like Molecules

The mathematical problem of enumerating the number of connected domains that can be drawn on a plane square lattice is studied by several methods, and results that are believed to be very nearly correct for large domains are obtained, while a slightly modified version of the problem can be solved in closed form. Comparison of the various methods tried leads to conclusions which, besides their purely mathematical interest, have a bearing on a large number of physical problems, such as ferromagnetism, order-disorder in alloys, the theory of solutions, fusion and evaporation, the configuration of polymer molecules and gel formation. The comparison of the various methods also gives information on what may be expected of an approximate theory of a phase-transition.