{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,25]],"date-time":"2025-11-25T05:00:25Z","timestamp":1764046825637,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We investigate the numbers $d_k$ of all (isomorphism classes of) distributive lattices with $k$ elements, or, equivalently, of (unlabeled) posets with $k$ antichains. Closely related and useful for combinatorial identities and inequalities are the numbers $v_k$ of vertically indecomposable distributive lattices of size $k$. We present the explicit values of the numbers $d_k$ and $v_k$ for $k &lt; 50$ and prove the following exponential bounds: $$ 1.67^k &lt; v_k &lt; 2.33^k\\;\\;\\; {\\rm and}\\;\\;\\; 1.84^k &lt; d_k &lt; 2.39^k\\;(k\\ge k_0).$$  Important tools are (i) an algorithm coding all unlabeled distributive lattices of height $n$ and size $k$ by certain integer sequences $0=z_1\\le\\cdots\\le z_n\\le k-2$, and (ii) a \"canonical 2-decomposition\" of ordinally indecomposable posets into \"2-indecomposable\" canonical summands.<\/jats:p>","DOI":"10.37236\/1641","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:26:44Z","timestamp":1578709604000},"source":"Crossref","is-referenced-by-count":12,"title":["On the Number of Distributive Lattices"],"prefix":"10.37236","volume":"9","author":[{"given":"Marcel","family":"Ern\u00e9","sequence":"first","affiliation":[]},{"given":"Jobst","family":"Heitzig","sequence":"additional","affiliation":[]},{"given":"J\u00fcrgen","family":"Reinhold","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2002,4,1]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v9i1r24\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v9i1r24\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:14:53Z","timestamp":1579324493000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v9i1r24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,4,1]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2002,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1641","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2002,4,1]]},"article-number":"R24"}}