We consider a directed compact site lattice animal problem on the $d$-dimensional hypercubic lattice, and establish its equivalence with (i) the infinite-state Potts model and (ii) the enumeration of $\left(d-1\mathrm{}\right)$-dimensional restricted partitions of an integer. The directed compact lattice animal problem is solved exactly in $d=2,3\mathrm{}$ using known solutions of the enumeration problem. The maximum number of lattice animals of size $n$ grows as $\exp \left({\mathrm{cn}}^{\left(d-1\mathrm{}\right)/d}\right)$. Also, the infinite-state Potts model solution leads to a conjectured limiting form for the generating function of restricted partitions for $d>\mathrm{}3\mathrm{}$, the latter an unsolved problem in number theory.

• Received 14 April 1995

DOI:http://dx.doi.org/10.1103/PhysRevLett.76.173