Directed Compact Lattice Animals, Restricted Partitions of an Integer, and the Infinite-State Potts Model

Abstract

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 $\left(d\ensuremath{-}1\right)$-dimensional restricted partitions of an integer. The directed compact lattice animal problem is solved exactly in $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2,3$ using known solutions of the enumeration problem. The maximum number of lattice animals of size $n$ grows as $\mathrm{exp}({\mathrm{cn}}^{\left(d\ensuremath{-}1\right)/d})$. Also, the infinite-state Potts model solution leads to a conjectured limiting form for the generating function of restricted partitions for $d>3$, the latter an unsolved problem in number theory.