  A vertical 2-sum of a two-coatom lattice $L$ and a two-atom lattice $U$ is
obtained by removing the top of $L$ and the bottom of $U$, and identifying the
coatoms of $L$ with the atoms of $U$. This operation creates one or two
nonisomorphic lattices depending on the symmetry case. Here the symmetry cases
are analyzed, and a recurrence relation is presented that expresses the number
of such vertical 2-sums in some family of interest, up to isomorphism.
Nonisomorphic, vertically indecomposable modular and distributive lattices are
counted and classified up to 35 and 60 elements respectively. Asymptotically
their numbers are shown to be at least $\Omega(2.3122^n)$ and
$\Omega(1.7250^n)$, where $n$ is the number of elements. The number of
semimodular lattices is shown to grow faster than any exponential in $n$.
