{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:28Z","timestamp":1753893808798,"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>Applying enumeration of sparse set partitions, we show that the number of set systems $H\\subset\\exp(\\{1,2,\\dots,n\\})$ such that $\\emptyset\\notin H$, $\\sum_{E\\in H} |E|=n$ and $\\bigcup_{E\\in H} E=\\{1,2,\\dots,m\\}$, $m\\le n$, equals $(1\/\\log(2)+o(1))^nb_n$ where $b_n$ is the $n$-th Bell number. The same asymptotics holds if $H$ may be a multiset.  If the vertex degrees in $H$ are restricted to be at most $k$, the asymptotics is $(1\/\\alpha_k+o(1))^nb_n$ where $\\alpha_k$ is the unique root of $\\sum_{i=1}^k x^i\/i!-1$ in $(0,1]$.<\/jats:p>","DOI":"10.37236\/1908","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:19:11Z","timestamp":1578719951000},"source":"Crossref","is-referenced-by-count":4,"title":["Counting Set Systems by Weight"],"prefix":"10.37236","volume":"12","author":[{"given":"Martin","family":"Klazar","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2005,2,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r11\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r11\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:51:05Z","timestamp":1579323065000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v12i1r11"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,2,14]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2005,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1908","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2005,2,14]]},"article-number":"R11"}}