Configuration space partitioning in tilings of a bounded region of the plane

Given a finite collection of two-dimensional tile types, we study the tiling of a rectangular region of the plane when the available tile types are all rectangular. Unlike the case of tiling the whole, unbounded plane, the additional complications imposed by the boundary conditions tend to constrain progress to mostly indirect results, such as recurrence relations. The tile types we use are squares, dominoes and straight tetraminoes, in which case not even recurrence relations are available. We seek to characterize this complex system through some fundamental physical quantities and follow two parallel tracks: One fully analytical for what seems to be the most complex special case still amenable to such approach, the other based on the Wang–Landau method for state-density estimation. Given a simple energy function based solely on tile contacts, we have found either approach to lead to illuminating depictions of entropy, temperature, and above all partitions of the configuration space. A configuration, in this context, refers to how many tiles of each type are used. We have found that certain partitions help bind together different aspects of the system in question and conjecture that future applications will benefit from the possibilities they afford.