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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 \u03a9(2.3122n<\/jats:italic><\/jats:sup>) and \u03a9(1.7250n<\/jats:italic><\/jats:sup>), where n<\/jats:italic> is the number of elements. The number of semimodular lattices is shown to grow faster than any exponential in n<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s11083-021-09569-0","type":"journal-article","created":{"date-parts":[[2021,5,25]],"date-time":"2021-05-25T11:03:13Z","timestamp":1621940593000},"page":"113-141","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Cartesian Lattice Counting by the Vertical 2-sum"],"prefix":"10.1007","volume":"39","author":[{"ORCID":"http:\/\/orcid.org\/0000-0003-4859-1463","authenticated-orcid":false,"given":"Jukka","family":"Kohonen","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,5,25]]},"reference":[{"key":"9569_CR1","doi-asserted-by":"publisher","first-page":"Article #R24","DOI":"10.37236\/1641","volume":"9","author":"M Ern\u00e9","year":"2002","unstructured":"Ern\u00e9, M., Heitzig, J, Reinhold, J: On the number of distributive lattices. Electron. 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