Exact Solution of a Directed-Site Animals-Enumeration Problem in Three Dimensions

An exact equivalence is established between the directed-site animals-enumeration (DSAE) problem in $d$ dimensions and a crystal-growth model defined on the same lattice. In special cases, the latter reduces to the calculation of the free energy of a ($d-1\mathrm{}$)-dimensional lattice gas with extended hard cores. The author solves a DSAE problem with $d=3\mathrm{}$ exactly by using its equivalence to the hard-hexagon problem, and shows that the exponent $\theta =\frac{5}{6}$. A new and simpler solution to the DSAE problem on the square and triangular lattices is given.