{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:20:48Z","timestamp":1759335648317,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Chen, Deng, Du, Stanley, and Yan introduced the notion of $k$-crossings and $k$-nestings for set partitions, and proved that the sizes of the largest $k$-crossings and $k$-nestings in the partitions of an $n$-set possess a symmetric joint distribution. This work considers a generalization of these results to set partitions whose arcs are labeled by an $r$-element set (which we call $r$-colored set partitions). In this context, a $k$-crossing or $k$-nesting is a sequence of arcs, all with the same color, which form a $k$-crossing or $k$-nesting in the usual sense. After showing that the sizes of the largest crossings and nestings in colored set partitions likewise have a symmetric joint distribution, we consider several related enumeration problems. We prove that $r$-colored set partitions with no crossing arcs of the same color are in bijection with certain paths in $\\mathbb{N}^r$, generalizing the correspondence between noncrossing (uncolored) set partitions and 2-Motzkin paths. Combining this with recent work of Bousquet-M\u00e9lou and Mishna affords a proof that the sequence counting noncrossing 2-colored set partitions is P-recursive. We also discuss how our methods extend to several variations of colored set partitions with analogous notions of crossings and nestings.<\/jats:p>","DOI":"10.37236\/3163","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:21:17Z","timestamp":1578705677000},"source":"Crossref","is-referenced-by-count":3,"title":["Crossings and Nestings in Colored Set Partitions"],"prefix":"10.37236","volume":"20","author":[{"given":"Eric","family":"Marberg","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,10,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i4p6\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i4p6\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:12:06Z","timestamp":1579259526000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i4p6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,10,21]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2013,10,14]]}},"URL":"https:\/\/doi.org\/10.37236\/3163","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,10,21]]},"article-number":"P6"}}