{"docId":2023,"paperId":1327,"url":"https:\/\/dmtcs.episciences.org\/1327","doi":"10.46298\/dmtcs.1327","journalName":"Discrete Mathematics & Theoretical Computer Science","issn":"","eissn":"1365-8050","volume":[{"vid":155,"name":"Vol. 18 no. 2, Permutation Patterns 2015"}],"section":[{"sid":61,"title":"Permutation Patterns","description":[]}],"repositoryName":"arXiv","repositoryIdentifier":"1511.00192","repositoryVersion":4,"repositoryLink":"https:\/\/arxiv.org\/abs\/1511.00192v4","dateSubmitted":"2016-09-07 09:49:48","dateAccepted":"2016-09-07 09:51:56","datePublished":"2016-09-07 09:52:25","titles":["Pattern avoidance for set partitions \\`a la Klazar"],"authors":["Bloom, Jonathan","Saracino, Dan"],"abstracts":["In 2000 Klazar introduced a new notion of pattern avoidance in the context of set partitions of $[n]=\\{1,\\ldots, n\\}$. The purpose of the present paper is to undertake a study of the concept of Wilf-equivalence based on Klazar's notion. We determine all Wilf-equivalences for partitions with exactly two blocks, one of which is a singleton block, and we conjecture that, for $n\\geq 4$, these are all the Wilf-equivalences except for those arising from complementation. If $\\tau$ is a partition of $[k]$ and $\\Pi_n(\\tau)$ denotes the set of all partitions of $[n]$ that avoid $\\tau$, we establish inequalities between $|\\Pi_n(\\tau_1)|$ and $|\\Pi_n(\\tau_2)|$ for several choices of $\\tau_1$ and $\\tau_2$, and we prove that if $\\tau_2$ is the partition of $[k]$ with only one block, then $|\\Pi_n(\\tau_1)| <|\\Pi_n(\\tau_2)|$ for all $n>k$ and all partitions $\\tau_1$ of $[k]$ with exactly two blocks. We conjecture that this result holds for all partitions $\\tau_1$ of $[k]$. Finally, we enumerate $\\Pi_n(\\tau)$ for all partitions $\\tau$ of $[4]$.","Comment: 21 pages"],"keywords":["Mathematics - Combinatorics","05A18"]}