Asymptotic results for the number of multidimensional partitions of an integer and directed compact lattice animals
D. P. Bhatia, M. A. Prasad, D. Arora · Journal of Physics A Mathematical and General · 1997
There is an exact one-to-one correspondence between the number of (d - 1)-dimensional partitions of an integer and the number of directed compact lattice animals in d dimensions. Using enumeration techniques, we obtain upper and lower bounds for the number of multidimensional partitions (both restricted and unrestricted). We show that asymptotically the number of unrestricted (d - 1)-dimensional partitions of an integer n goes as exp . We also show that for restricted partitions in (d - 1) dimensions (with j dimensions finite, 0 < j < d-1), this number goes as , where is the extent of the lattice along the kth axis.