{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T10:06:31Z","timestamp":1770285991788,"version":"3.49.0"},"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>A (multi)hypergraph ${\\cal H}$ with vertices in ${\\bf N}$ contains a permutation $p=a_1a_2\\ldots a_k$ of $1, 2, \\ldots, k$ if one can reduce ${\\cal H}$ by omitting vertices from the edges so that the resulting hypergraph is isomorphic, via an increasing mapping, to ${\\cal H}_p=(\\{i, k+a_i\\}:\\ i=1, \\ldots, k)$. We formulate six conjectures stating that if ${\\cal H}$ has $n$ vertices and does not contain $p$ then the size of ${\\cal H}$ is $O(n)$ and the number of such ${\\cal H}$s is $O(c^n)$. The latter part generalizes the Stanley\u2013Wilf conjecture on permutations. Using generalized Davenport\u2013Schinzel sequences, we prove the conjectures with weaker bounds $O(n\\beta(n))$ and $O(\\beta(n)^n)$, where $\\beta(n)\\rightarrow\\infty$ very slowly. We prove the conjectures fully if $p$ first increases and then decreases or if $p^{-1}$ decreases and then  increases. For the cases $p=12$ (noncrossing structures) and $p=21$ (nonnested structures) we give many precise enumerative and extremal results, both for graphs and hypergraphs.<\/jats:p>","DOI":"10.37236\/1512","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:04:31Z","timestamp":1578708271000},"source":"Crossref","is-referenced-by-count":17,"title":["Counting Pattern-free Set Partitions II: Noncrossing and Other Hypergraphs"],"prefix":"10.37236","volume":"7","author":[{"given":"Martin","family":"Klazar","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2000,5,23]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r34\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r34\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:23:00Z","timestamp":1579324980000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v7i1r34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,5,23]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2000,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1512","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,5,23]]},"article-number":"R34"}}