Configurational entropy of codimension-one tilings and directed membranes

The calculation of random tiling configurational entropy amounts to an enumeration of partitions. A geometrical description of the configuration space is given in terms of integral points in a high-dimensional space, and the entropy is deduced from the integral volume of a convex polytope. In some cases the latter volume can be expressed in a compact multiplicative formula, and in all cases in terms of binomial series, the origin of which is given a geometrical meaning. Our results mainly concern codimension-one tilings, but can also be extended to higher codimension tilings. We also discuss the link between free-boundary- and fixed-boundary-condition problems.