  We study the number of $k \times r$ plane partitions, weighted on the sum of
the first row. Using Erhart reciprocity, we prove an identity for the
generating function. For the special case $k=1$ this result follows from the
classical theory of partitions, and for $k=2$ it was proved in
Andersson-Bhowmik with another method. We give an explicit formula in terms of
Young tableaux, and study the corresponding zeta-function. We give an
application on the average orders of towers of abelian groups. In particular we
prove that the number of isomorphism classes of ``subgroups of subgroups of ...
($k-1$ times) ... of abelian groups'' of order at most $N$ is asymptotic to
$c_k N (\log N)^{k-1}$. This generalises results from Erd{\H o}s-Szekeres and
Andersson-Bhowmik where the corresponding result was proved for $k=1$ and
$k=2$.
