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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1308.2666","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-08-12T19:58:26Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"df3756a6f038d0084f5df90405147dc0140c95cd4bc03ee6e7d204f8ee6b0a0e","abstract_canon_sha256":"c3ced13da54505b3b5d950d0b19a2803bd2ecfa5a05d537c031e8ad35d345302"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:47:05.717597Z","signature_b64":"6nBB/C50FAEjXrzEFUKIVHpni+NDTeFkNh9+JXR2onxYMsshcJ1QhN/wjjx5WaMBCm5WPDHgNYk+WMnrWkXsAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ffe7da78d621e2ff2dfecc47d271e07de74a6eac5db5279d63b073f9e8b0e366","last_reissued_at":"2026-05-18T02:47:05.717144Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:47:05.717144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1308.2666","created_at":"2026-05-18T02:47:05.717224+00:00"},{"alias_kind":"arxiv_version","alias_value":"1308.2666v5","created_at":"2026-05-18T02:47:05.717224+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.2666","created_at":"2026-05-18T02:47:05.717224+00:00"},{"alias_kind":"pith_short_12","alias_value":"77T5U6GWEHRP","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_16","alias_value":"77T5U6GWEHRP6LP6","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_8","alias_value":"77T5U6GW","created_at":"2026-05-18T12:27:36.564083+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/77T5U6GWEHRP6LP6ZRD5E4PAPX","json":"https://pith.science/pith/77T5U6GWEHRP6LP6ZRD5E4PAPX.json","graph_json":"https://pith.science/api/pith-number/77T5U6GWEHRP6LP6ZRD5E4PAPX/graph.json","events_json":"https://pith.science/api/pith-number/77T5U6GWEHRP6LP6ZRD5E4PAPX/events.json","paper":"https://pith.science/paper/77T5U6GW"},"agent_actions":{"view_html":"https://pith.science/pith/77T5U6GWEHRP6LP6ZRD5E4PAPX","download_json":"https://pith.science/pith/77T5U6GWEHRP6LP6ZRD5E4PAPX.json","view_paper":"https://pith.science/paper/77T5U6GW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1308.2666&json=true","fetch_graph":"https://pith.science/api/pith-number/77T5U6GWEHRP6LP6ZRD5E4PAPX/graph.json","fetch_events":"https://pith.science/api/pith-number/77T5U6GWEHRP6LP6ZRD5E4PAPX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/77T5U6GWEHRP6LP6ZRD5E4PAPX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/77T5U6GWEHRP6LP6ZRD5E4PAPX/action/storage_attestation","attest_author":"https://pith.science/pith/77T5U6GWEHRP6LP6ZRD5E4PAPX/action/author_attestation","sign_citation":"https://pith.science/pith/77T5U6GWEHRP6LP6ZRD5E4PAPX/action/citation_signature","submit_replication":"https://pith.science/pith/77T5U6GWEHRP6LP6ZRD5E4PAPX/action/replication_record"}},"created_at":"2026-05-18T02:47:05.717224+00:00","updated_at":"2026-05-18T02:47:05.717224+00:00"}