Activity-space systems: four isomorphisms

Four isomorphisms linking architectural problems to combinatorial mathematics answer four questions: How many ways can a given population be partitioned into a given schedule of groups? What is the most probable way that a given population will divide into a given number of groups? How many ways can a set of activities be allocated to a set of spaces? How many ways can subsets of a set of activities be allocated to a set of spaces? Further unsolved questions are also discussed.