{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:KDIN6WQS23W6WXEW7UFWWJRB6W","short_pith_number":"pith:KDIN6WQS","canonical_record":{"source":{"id":"1903.06317","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-15T01:42:50Z","cross_cats_sorted":["cs.DM","math.PR"],"title_canon_sha256":"b1e3d8c95b144d4b55f45195fd4ec9a2c1d311d92ef8ca95b4ec53d1ef12d643","abstract_canon_sha256":"130b1cc379f2f6cdea88160cecadf7efb5ecaf1eb3d83a99bb81b95cc0d59d4f"},"schema_version":"1.0"},"canonical_sha256":"50d0df5a12d6edeb5c96fd0b6b2621f583cb5253f37acd404f853fd40a55fd71","source":{"kind":"arxiv","id":"1903.06317","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"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:KDIN6WQS23W6WXEW7UFWWJRB6W","target":"record","payload":{"canonical_record":{"source":{"id":"1903.06317","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-15T01:42:50Z","cross_cats_sorted":["cs.DM","math.PR"],"title_canon_sha256":"b1e3d8c95b144d4b55f45195fd4ec9a2c1d311d92ef8ca95b4ec53d1ef12d643","abstract_canon_sha256":"130b1cc379f2f6cdea88160cecadf7efb5ecaf1eb3d83a99bb81b95cc0d59d4f"},"schema_version":"1.0"},"canonical_sha256":"50d0df5a12d6edeb5c96fd0b6b2621f583cb5253f37acd404f853fd40a55fd71","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:11.150222Z","signature_b64":"bLBGDRhfhyh/Zhb+lIGQn7m8Taa78L6+znsffqc6Yt2azIyr5rvMLWdkpDxQGl+YznWBEJPW0Gbf2fAch/TQBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50d0df5a12d6edeb5c96fd0b6b2621f583cb5253f37acd404f853fd40a55fd71","last_reissued_at":"2026-05-17T23:51:11.149703Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:11.149703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1903.06317","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:51:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vix8Xrdd/cmct7IdAZaoF4sT0Lhm2UeRa8XOVu4zkVFp3wZvM1yqf4ZnXABMqRqM2vKeHRndi7NGeDhT6xElCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T18:25:23.608514Z"},"content_sha256":"c212d17bcb51b90578d8345c1a003c836ff21560635d378da12a4810e3727084","schema_version":"1.0","event_id":"sha256:c212d17bcb51b90578d8345c1a003c836ff21560635d378da12a4810e3727084"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:KDIN6WQS23W6WXEW7UFWWJRB6W","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Limits of Sums for Binomial and Eulerian Numbers and their Associated Distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.PR"],"primary_cat":"math.CO","authors_text":"Meng Li, Ron Goldman","submitted_at":"2019-03-15T01:42:50Z","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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.06317","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:51:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2oTrFtQDao8owgz6W8kCnkvQZdRNlQG94T8qNflSlybE2YsZHwLI4VcjuHYi02ck0BP2WGz+/jvq0CjLh50FAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T18:25:23.608850Z"},"content_sha256":"77b5e0ba5168a0be4a168690c10eeb67f9e5f689d69e3611c732b40c88936646","schema_version":"1.0","event_id":"sha256:77b5e0ba5168a0be4a168690c10eeb67f9e5f689d69e3611c732b40c88936646"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KDIN6WQS23W6WXEW7UFWWJRB6W/bundle.json","state_url":"https://pith.science/pith/KDIN6WQS23W6WXEW7UFWWJRB6W/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KDIN6WQS23W6WXEW7UFWWJRB6W/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-27T18:25:23Z","links":{"resolver":"https://pith.science/pith/KDIN6WQS23W6WXEW7UFWWJRB6W","bundle":"https://pith.science/pith/KDIN6WQS23W6WXEW7UFWWJRB6W/bundle.json","state":"https://pith.science/pith/KDIN6WQS23W6WXEW7UFWWJRB6W/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KDIN6WQS23W6WXEW7UFWWJRB6W/bundle.json"},"state":{"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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nytHqi6X8FhxuJXynUfCGKjzw0g3lv1Xnc1LrnxfwYt8PSzaS/JWg5tkKCAmEzp/5rqYep0nA5Rg1WzckerpCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T18:25:23.610703Z","bundle_sha256":"73e6c6853c659bf3dcdc4f6a9a3c0151836ea092874d0d4a16468479eb715de2"}}