  We study a one-parameter family ($\ell=1,2,3,\ldots$) of configurations that
are square-ice analogues of plane partitions. Using an algorithm due to Bratley
and McKay, we carry out exact enumerations in order to study their asymptotic
behaviour and establish, via Monte Carlo simulations as well as explicit
bounds, that the asymptotic behaviour is similar to that of plane partitions.
We finally carry out a series analysis and provide independent estimates for
the asymptotic behaviour.
