{"paper":{"title":"On xD-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sen-Peng Eu, Tsai-Lien Wong, Tung-Shan Fu, Yu-Chang Liang","submitted_at":"2017-01-03T07:51:41Z","abstract_excerpt":"This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal order problem in the Weyl algebra $W=\\langle x,D|Dx-xD=1\\rangle$. Any word $\\omega\\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\\omega=x^{m-n}\\sum_{k\\ge 0} {{\\omega}\\brace {k}} x^{k}D^{k}$, where ${{\\omega}\\brace {k}}$ is known as the Stirling number of the second kind for the word $\\omega$. This study considers the expansions of restricted words $\\omega$ in $W$ over the sequences $\\{(xD)^{k}\\}_{k\\ge 0}$ and $\\{xD^{k}x^{k-1}\\}_{k\\g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.00600","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"}