{"paper":{"title":"Partitions, rooks, and symmetric functions in noncommuting variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bruce E. Sagan, Mahir Bilen Can","submitted_at":"2010-08-17T19:23:18Z","abstract_excerpt":"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 $\\cE_n\\sbe\\Pi_n$ be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, $\\cT_{n-1}$. Given $\\pi\\in\\Pi_m$ and $\\si\\in\\Pi_n$, define their {\\it slash product\\/} to be $\\pi|\\si=\\pi\\cup(\\si+m)\\in\\Pi_{m+n}$ where $\\si+m$ is the partition obtained by adding $m$ to every element of every block of $\\si$. Call $\\tau$ {\\it atomic\\/} if it c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.2950","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}