{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T09:51:52Z","timestamp":1770285112822,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize $3$-crossings and $3$-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards.We enumerate $312$-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the $321$-avoiding (i.e., $3$-noncrossing) case. Our approach provides a more direct proof of a formula of B\u00f3na for the number of $1342$-avoiding permutations. We also give a bijective proof of the shape-Wilf-equivalence of the patterns $321$ and $213$ which greatly simplifies existing proofs by Backelin-West-Xin and Jel\u00ednek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.<\/jats:p>","DOI":"10.37236\/2976","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:13:38Z","timestamp":1578719618000},"source":"Crossref","is-referenced-by-count":13,"title":["Pattern Avoidance in Matchings and Partitions"],"prefix":"10.37236","volume":"20","author":[{"given":"Jonathan","family":"Bloom","sequence":"first","affiliation":[]},{"given":"Sergi","family":"Elizalde","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,4,9]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p5\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p5\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:22:59Z","timestamp":1579260179000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i2p5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,4,9]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2013,4,9]]}},"URL":"https:\/\/doi.org\/10.37236\/2976","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,4,9]]},"article-number":"P5"}}