DIRECTED COMPACT LATTICE ANIMALS, RESTRICTED PARTITION OF AN INTEGER,AND THE INFINITE-STATE POTTS-MODEL

We consider a directed compact site lattice animal problem on the d-di mensional hypercubic lattice, and establish its equivalence with (i) t he infinite-state Potts model and (ii) the enumeration of(d-1)-dimensi onal restricted partitions of an integer. The directed compact lattice animal problem is solved exactly in d = 2,3 using known solutions of the enumeration problem. The maximum number of lattice animals of size n grows as exp(cn((d-1)/d)). Also, the infinite-state Potts. model so lution leads to a conjectured limiting form for the generating functio n of restricted partitions for d > 3, the latter an unsolved problem i n number theory.