  We consider configurational statistics of three-dimensional heaps of $N$
pieces ($N\gg 1$) on a simple cubic lattice in a large 3D bounding box of base
$n \times n$, and calculate the growth rate, $\Lambda(n)$, of the corresponding
partition function, $Z_N\sim N^{\theta}[\Lambda(n)]^N$, at $n\gg 1$. Our
computations rely on a theorem of G.X. Viennot \cite{viennot-rev}, which
connects the generating function of a $(D+1)$-dimensional heap of pieces to the
generating function of projection of these pieces onto a $D$-dimensional
subspace. The growth rate of a heap of cubic blocks, which cannot touch each
other by vertical faces, is thus related to the position of zeros of the
partition function describing 2D lattice gas of hard squares. We study the
corresponding partition function exactly at low densities on finite $n\times n$
lattice of arbitrary $n$, and extrapolate its behavior to the jamming
transition density. This allows us to estimate the limiting growth rate,
$\Lambda =\lim_{n\to\infty}\Lambda(n)\approx 9.5$. The same method works for
any underlying 2D lattice and for various shapes of pieces: flat vertical
squares, mapped to an ensemble of repulsive dimers, dominoes mapped to an
ensemble of rectangles with hard-core repulsion, etc.
