{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:KDIN6WQS23W6WXEW7UFWWJRB6W","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"130b1cc379f2f6cdea88160cecadf7efb5ecaf1eb3d83a99bb81b95cc0d59d4f","cross_cats_sorted":["cs.DM","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-15T01:42:50Z","title_canon_sha256":"b1e3d8c95b144d4b55f45195fd4ec9a2c1d311d92ef8ca95b4ec53d1ef12d643"},"schema_version":"1.0","source":{"id":"1903.06317","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.06317","created_at":"2026-05-17T23:51:11Z"},{"alias_kind":"arxiv_version","alias_value":"1903.06317v1","created_at":"2026-05-17T23:51:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.06317","created_at":"2026-05-17T23:51:11Z"},{"alias_kind":"pith_short_12","alias_value":"KDIN6WQS23W6","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"KDIN6WQS23W6WXEW","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"KDIN6WQS","created_at":"2026-05-18T12:33:21Z"}],"graph_snapshots":[{"event_id":"sha256:77b5e0ba5168a0be4a168690c10eeb67f9e5f689d69e3611c732b40c88936646","target":"graph","created_at":"2026-05-17T23:51:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We provide a unified, probabilistic approach using renewal theory to derive some novel limits of sums for the normalized binomial coefficients and for the normalized Eulerian numbers. We also investigate some corresponding results for their associated distributions -- the binomial distributions for the binomial coefficients and the Irwin-Hall distributions (uniform B-splines) for the Eulerian numbers.","authors_text":"Meng Li, Ron Goldman","cross_cats":["cs.DM","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-15T01:42:50Z","title":"Limits of Sums for Binomial and Eulerian Numbers and their Associated Distributions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.06317","kind":"arxiv","version":1},"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:c212d17bcb51b90578d8345c1a003c836ff21560635d378da12a4810e3727084","target":"record","created_at":"2026-05-17T23:51:11Z","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":"130b1cc379f2f6cdea88160cecadf7efb5ecaf1eb3d83a99bb81b95cc0d59d4f","cross_cats_sorted":["cs.DM","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-15T01:42:50Z","title_canon_sha256":"b1e3d8c95b144d4b55f45195fd4ec9a2c1d311d92ef8ca95b4ec53d1ef12d643"},"schema_version":"1.0","source":{"id":"1903.06317","kind":"arxiv","version":1}},"canonical_sha256":"50d0df5a12d6edeb5c96fd0b6b2621f583cb5253f37acd404f853fd40a55fd71","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50d0df5a12d6edeb5c96fd0b6b2621f583cb5253f37acd404f853fd40a55fd71","first_computed_at":"2026-05-17T23:51:11.149703Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:11.149703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bLBGDRhfhyh/Zhb+lIGQn7m8Taa78L6+znsffqc6Yt2azIyr5rvMLWdkpDxQGl+YznWBEJPW0Gbf2fAch/TQBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:11.150222Z","signed_message":"canonical_sha256_bytes"},"source_id":"1903.06317","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c212d17bcb51b90578d8345c1a003c836ff21560635d378da12a4810e3727084","sha256:77b5e0ba5168a0be4a168690c10eeb67f9e5f689d69e3611c732b40c88936646"],"state_sha256":"b22ec9e7ad7165f7c7a7b8288a83bcdc1c4b5457ac064313c7cb9810c2d725a4"}