{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T04:10:24Z","timestamp":1775535024293,"version":"3.50.1"},"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>Let $\\Pi_n$ denote the set of all set partitions of $\\{1,2,\\ldots,n\\}$.  We consider two subsets of $\\Pi_n$, one connected to rook theory and one associated with symmetric functions in noncommuting variables.  Let ${\\cal E}_n\\subseteq\\Pi_n$ be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, ${\\cal T}_{n-1}$.  Given $\\pi\\in\\Pi_m$ and $\\sigma\\in\\Pi_n$, define their slash product to be $\\pi|\\sigma=\\pi\\cup(\\sigma+m)\\in\\Pi_{m+n}$ where $\\sigma+m$ is the partition obtained by adding $m$ to every element of every block of $\\sigma$.  Call $\\tau$ atomic if it can not be written as a nontrivial slash product and let ${\\cal A}_n\\subseteq\\Pi_n$ denote the subset of atomic partitions.  Atomic partitions were first defined by Bergeron, Hohlweg, Rosas, and Zabrocki during their study of $NCSym$, the symmetric functions in noncommuting variables.  We show that, despite their very different definitions, ${\\cal E}_n={\\cal A}_n$ for all $n\\ge0$.  Furthermore, we put an algebra structure on the formal vector space generated by all rook placements on upper triangular boards which makes it isomorphic to $NCSym$.  We end with some remarks.<\/jats:p>","DOI":"10.37236\/1999","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:38:07Z","timestamp":1578695887000},"source":"Crossref","is-referenced-by-count":4,"title":["Partitions, Rooks, and Symmetric Functions in Noncommuting Variables"],"prefix":"10.37236","volume":"18","author":[{"given":"Mahir Bilen","family":"Can","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bruce E.","family":"Sagan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,1,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i2p3\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i2p3\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T18:17:41Z","timestamp":1579285061000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i2p3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1,24]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2011,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1999","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1,24]]},"article-number":"P3"}}