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 nonisomorphic vertical 2-sums in some desired family of graded lattices. 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 Ω(2.3122ⁿ) and Ω(1.7250ⁿ), where n is the number of elements. The number of semimodular lattices is shown to grow faster than any exponential in n.

通过去除L的顶部和U的底部，并用U的原子标识L的覆盖层，可以得到两个原子的晶格L和两个原子的晶格U的垂直2和。此操作根据对称情况创建一个或两个非同构晶格。在这里，对对称情况进行了分析，并给出了一个递归关系，该递归关系表示某些所需的渐变格族中非同构垂直2-和的数量。非同构，垂直不可分解的模块化晶格和分布晶格分别进行计数和分类，最多可包含35个元素和60个元素。渐近地显示它们的数量至少为Ω（2.3122 ⁿ）和Ω（1.7250 ⁿ），其中n是元素数。显示出半模晶格的数量增长快于n中的任何指数增长。