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These bounds are useful for values of $m \\geq n - O(\\sqrt{n})$. An application of our Theorem 5 yields, for example, \\[ s(10^{12},\\ 10^{12}-2\\times 10^6)/10^{35664464} \\in [ 1.87669, 1.876982 ], \\] \\[ S(10^{12},\\ 10^{12}-2\\times 10^6)/10^{35664463} \\in [ 1.30121, 1.306975 ]. \\] The bounds are obtained via Chen-Stein Poisson approximation, using an interpretation of Stirling numbers as the number of ways of placing non-attacking rooks on a chess boar","authors_text":"Richard Arratia, Stephen DeSalvo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-04-11T05:51:39Z","title":"Completely effective error bounds for Stirling Numbers of the first and second kind via Poisson Approximation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.3007","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6eee0fa3c42596ed45004c54f791a000d6b1c0cfd7de1a0c0e679513a82e3921","target":"record","created_at":"2026-05-18T01:04:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c518eb025c29b36f3eb0967cd41dd70a2191ee6a19408617e9e45ab2256757f3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-04-11T05:51:39Z","title_canon_sha256":"07b2b27db3835254b8bc28083ca73c95a36b2bc15164ecc679e8a23a95a1f285"},"schema_version":"1.0","source":{"id":"1404.3007","kind":"arxiv","version":4}},"canonical_sha256":"d15989da5e3821cb2d24a957c4e46813036a8744de8a293fdf6a2aa947a6684f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d15989da5e3821cb2d24a957c4e46813036a8744de8a293fdf6a2aa947a6684f","first_computed_at":"2026-05-18T01:04:54.268669Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:54.268669Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NeATLDGr/QeLsrEqqfzFQ77akivkWOnw3ZyJYOanxycy+/9D3XjuYTExqC80Vkse1gmozncYVgGQjj7ii52RBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:54.269255Z","signed_message":"canonical_sha256_bytes"},"source_id":"1404.3007","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6eee0fa3c42596ed45004c54f791a000d6b1c0cfd7de1a0c0e679513a82e3921","sha256:87a49c5f47f8349819a91987a9e41af68fdbb3f2766fb6c32fef2d7555340226"],"state_sha256":"10dd58baad3b3505e8e94c861cc3a3850b4e6d30bee285b1502c9bbcb864a675"}