{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:CJVYQYKWR7GIUO47AYUHGBOBCN","short_pith_number":"pith:CJVYQYKW","canonical_record":{"source":{"id":"1604.04028","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-14T03:57:56Z","cross_cats_sorted":[],"title_canon_sha256":"58df6066fafcfa7681612203244a93c70f6cace1325fe41ffa74563d496f7181","abstract_canon_sha256":"4dba9a7392e8010c48df336b7018e0be39fad6e6ecde7c4834d77e93d10dbf53"},"schema_version":"1.0"},"canonical_sha256":"126b8861568fcc8a3b9f06287305c11369a5a84eb2b86c7c74dedb09f3f70101","source":{"kind":"arxiv","id":"1604.04028","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.04028","created_at":"2026-05-18T01:17:06Z"},{"alias_kind":"arxiv_version","alias_value":"1604.04028v1","created_at":"2026-05-18T01:17:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.04028","created_at":"2026-05-18T01:17:06Z"},{"alias_kind":"pith_short_12","alias_value":"CJVYQYKWR7GI","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CJVYQYKWR7GIUO47","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CJVYQYKW","created_at":"2026-05-18T12:30:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:CJVYQYKWR7GIUO47AYUHGBOBCN","target":"record","payload":{"canonical_record":{"source":{"id":"1604.04028","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-14T03:57:56Z","cross_cats_sorted":[],"title_canon_sha256":"58df6066fafcfa7681612203244a93c70f6cace1325fe41ffa74563d496f7181","abstract_canon_sha256":"4dba9a7392e8010c48df336b7018e0be39fad6e6ecde7c4834d77e93d10dbf53"},"schema_version":"1.0"},"canonical_sha256":"126b8861568fcc8a3b9f06287305c11369a5a84eb2b86c7c74dedb09f3f70101","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:06.903589Z","signature_b64":"wgIrkJCF86hEoGShUL88xglMedAtDahxoWQ8EUvQo6SCgQygKnpeI7MyObyP91NzqMC1Flj6dRRJcKt06rQBAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"126b8861568fcc8a3b9f06287305c11369a5a84eb2b86c7c74dedb09f3f70101","last_reissued_at":"2026-05-18T01:17:06.902851Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:06.902851Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.04028","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":"2026-05-18T01:17:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HDj9HVsA5ehieMbfSz0amX7hxWKyVf/pb21BprMe7De+N+vTlFJINzUCva8wGEecu1GAolk7OYFdcapl4Vo0Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T05:21:39.387988Z"},"content_sha256":"8b1c5f4323dc8c9ff11ff6e45a1592e314f17286de39e220a9be4e5342d38afc","schema_version":"1.0","event_id":"sha256:8b1c5f4323dc8c9ff11ff6e45a1592e314f17286de39e220a9be4e5342d38afc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:CJVYQYKWR7GIUO47AYUHGBOBCN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Partitions with fixed largest hook length","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dazhao Tang, Shishuo Fu","submitted_at":"2016-04-14T03:57:56Z","abstract_excerpt":"Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub's analogue of Euler's Odd-Distinct partition theorem, derive a generalization in the spirit of Alder's conjecture, as well as a curious analogue of the first Rogers-Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler's pentagonal number theory in this setting, and connect it with the Rogers-Fine identity. We concludes with some congruence "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.04028","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":"2026-05-18T01:17:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/GtjirZrcR9zqYP1NyquW9vBB4miLSKOO43F9KsPg0M3+XQyvSoFoquoTJaBZC/tNf7tsOh1qJ38lLS6OxaUAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T05:21:39.388335Z"},"content_sha256":"1b32454f4b57ee2a4f49c3fe130aa755ab4e8069109d9115de3d2c7ccb2db76c","schema_version":"1.0","event_id":"sha256:1b32454f4b57ee2a4f49c3fe130aa755ab4e8069109d9115de3d2c7ccb2db76c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CJVYQYKWR7GIUO47AYUHGBOBCN/bundle.json","state_url":"https://pith.science/pith/CJVYQYKWR7GIUO47AYUHGBOBCN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CJVYQYKWR7GIUO47AYUHGBOBCN/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":"2026-07-05T05:21:39Z","links":{"resolver":"https://pith.science/pith/CJVYQYKWR7GIUO47AYUHGBOBCN","bundle":"https://pith.science/pith/CJVYQYKWR7GIUO47AYUHGBOBCN/bundle.json","state":"https://pith.science/pith/CJVYQYKWR7GIUO47AYUHGBOBCN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CJVYQYKWR7GIUO47AYUHGBOBCN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:CJVYQYKWR7GIUO47AYUHGBOBCN","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":"4dba9a7392e8010c48df336b7018e0be39fad6e6ecde7c4834d77e93d10dbf53","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-14T03:57:56Z","title_canon_sha256":"58df6066fafcfa7681612203244a93c70f6cace1325fe41ffa74563d496f7181"},"schema_version":"1.0","source":{"id":"1604.04028","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.04028","created_at":"2026-05-18T01:17:06Z"},{"alias_kind":"arxiv_version","alias_value":"1604.04028v1","created_at":"2026-05-18T01:17:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.04028","created_at":"2026-05-18T01:17:06Z"},{"alias_kind":"pith_short_12","alias_value":"CJVYQYKWR7GI","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CJVYQYKWR7GIUO47","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CJVYQYKW","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:1b32454f4b57ee2a4f49c3fe130aa755ab4e8069109d9115de3d2c7ccb2db76c","target":"graph","created_at":"2026-05-18T01:17:06Z","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":"Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub's analogue of Euler's Odd-Distinct partition theorem, derive a generalization in the spirit of Alder's conjecture, as well as a curious analogue of the first Rogers-Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler's pentagonal number theory in this setting, and connect it with the Rogers-Fine identity. We concludes with some congruence ","authors_text":"Dazhao Tang, Shishuo Fu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-14T03:57:56Z","title":"Partitions with fixed largest hook length"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.04028","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":"sha256:8b1c5f4323dc8c9ff11ff6e45a1592e314f17286de39e220a9be4e5342d38afc","target":"record","created_at":"2026-05-18T01:17:06Z","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":"4dba9a7392e8010c48df336b7018e0be39fad6e6ecde7c4834d77e93d10dbf53","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-14T03:57:56Z","title_canon_sha256":"58df6066fafcfa7681612203244a93c70f6cace1325fe41ffa74563d496f7181"},"schema_version":"1.0","source":{"id":"1604.04028","kind":"arxiv","version":1}},"canonical_sha256":"126b8861568fcc8a3b9f06287305c11369a5a84eb2b86c7c74dedb09f3f70101","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"126b8861568fcc8a3b9f06287305c11369a5a84eb2b86c7c74dedb09f3f70101","first_computed_at":"2026-05-18T01:17:06.902851Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:06.902851Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wgIrkJCF86hEoGShUL88xglMedAtDahxoWQ8EUvQo6SCgQygKnpeI7MyObyP91NzqMC1Flj6dRRJcKt06rQBAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:06.903589Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.04028","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8b1c5f4323dc8c9ff11ff6e45a1592e314f17286de39e220a9be4e5342d38afc","sha256:1b32454f4b57ee2a4f49c3fe130aa755ab4e8069109d9115de3d2c7ccb2db76c"],"state_sha256":"7edde8a72bac51dfa5f1ab125a9f5d63a293f2d6c57ed3d45f9e5784cb44c4a7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DTE/AgyT1j3a0Mhhwm0UMkZDeYhwqJKaEe7c4wJpuFu7T8MgN0qeAoRmUbTgyp8Dq7a7zdK3rxtSj1wyigZxAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T05:21:39.390478Z","bundle_sha256":"2df4ad36853b199bd990f2ea937ea4331d7fafac4519b269ac4b2a2275234464"}}