{"paper":{"title":"Combinatorially interpreting generalized Stirling numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"David Galvin, John Engbers, Justin Hilyard","submitted_at":"2013-08-12T19:58:26Z","abstract_excerpt":"Let $w$ be a word in alphabet $\\{x,D\\}$ with $m$ $x$'s and $n$ $D$'s. Interpreting \"$x$\" as multiplication by $x$, and \"$D$\" as differentiation with respect to $x$, the identity $wf(x) = x^{m-n}\\sum_k S_w(k) x^k D^k f(x)$, valid for any smooth function $f(x)$, defines a sequence $(S_w(k))_k$, the terms of which we refer to as the {\\em Stirling numbers (of the second kind)} of $w$. The nomenclature comes from the fact that when $w=(xD)^n$, we have $S_w(k)={n \\brace k}$, the ordinary Stirling number of the second kind.\n  Explicit expressions for, and identities satisfied by, the $S_w(k)$ have be"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.2666","kind":"arxiv","version":5},"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"}