{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:PFWF5GBKVQLWO6W5DMVJV4H6JA","short_pith_number":"pith:PFWF5GBK","canonical_record":{"source":{"id":"1208.2210","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-10T16:13:54Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"a7a200e0730358abc6972dcc0106d0aaa686bcd73fa8baf1ffa48bb53a9a50f5","abstract_canon_sha256":"c14c560dcc66fd7740a86c8951080d4364e1938c2c3c15d43f18ad5914e4d02d"},"schema_version":"1.0"},"canonical_sha256":"796c5e982aac17677add1b2a9af0fe4823607b2e573630e9bfbc4436f5e84615","source":{"kind":"arxiv","id":"1208.2210","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.2210","created_at":"2026-05-18T03:48:59Z"},{"alias_kind":"arxiv_version","alias_value":"1208.2210v1","created_at":"2026-05-18T03:48:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.2210","created_at":"2026-05-18T03:48:59Z"},{"alias_kind":"pith_short_12","alias_value":"PFWF5GBKVQLW","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"PFWF5GBKVQLWO6W5","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"PFWF5GBK","created_at":"2026-05-18T12:27:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:PFWF5GBKVQLWO6W5DMVJV4H6JA","target":"record","payload":{"canonical_record":{"source":{"id":"1208.2210","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-10T16:13:54Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"a7a200e0730358abc6972dcc0106d0aaa686bcd73fa8baf1ffa48bb53a9a50f5","abstract_canon_sha256":"c14c560dcc66fd7740a86c8951080d4364e1938c2c3c15d43f18ad5914e4d02d"},"schema_version":"1.0"},"canonical_sha256":"796c5e982aac17677add1b2a9af0fe4823607b2e573630e9bfbc4436f5e84615","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:48:59.873313Z","signature_b64":"kOR7+kdDtU4oO6hRfL3BQQj/sEdSwAztRbUCfWRf++ODXJvK/4z+qKtgBkJpL9LMZNVc7i8OztNftAykpTR6Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"796c5e982aac17677add1b2a9af0fe4823607b2e573630e9bfbc4436f5e84615","last_reissued_at":"2026-05-18T03:48:59.872419Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:48:59.872419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.2210","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":"2026-05-18T03:48:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vxUTze46+VU+jNQALiVIxGVUpkggkoemwcwDKzivmvH67iUhCf6mwNe16Fkpq7YCV/rXeBPQ+YmDKTBIsh0uBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T20:47:11.991654Z"},"content_sha256":"149e2f58feac09e5b5d4686dc8907302f75821aa5a63766231ede5460eb064ce","schema_version":"1.0","event_id":"sha256:149e2f58feac09e5b5d4686dc8907302f75821aa5a63766231ede5460eb064ce"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:PFWF5GBKVQLWO6W5DMVJV4H6JA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Number of Partitions with Designated Summands","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Erin Y. Y. Shen, Hai-Tao Jin, Kathy Q. Ji, William Y. C. Chen","submitted_at":"2012-08-10T16:13:54Z","abstract_excerpt":"Andrews, Lewis and Lovejoy introduced the partition function PD(n) as the number of partitions of $n$ with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that PD(3n+2) is divisible by 3. We obtain a Ramanujan type identity for the generating function of PD(3n+2) which implies the congruence of Andrews, Lewis and Lovejoy. For PD(3n), Andrews, Lewis and Lovejoy showed that the generating function can be expressed as an infinite product of powers of $(1-q^{2n+1})$ times a function $F(q^2)$. We find an explicit formula for $F(q^2)$, wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.2210","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":"2026-05-18T03:48:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vUfOMXid8RwACWGup8jdF4eLniA9PJoHaYiKYeO1fAW8w4THaIbpa5bmXPerCCRmaMUK99DnJFSGtlQngpvFDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T20:47:11.992309Z"},"content_sha256":"2c69bcea0b1460329e67318da169aee06b8143342fb1ed8deb26833fdc363f38","schema_version":"1.0","event_id":"sha256:2c69bcea0b1460329e67318da169aee06b8143342fb1ed8deb26833fdc363f38"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PFWF5GBKVQLWO6W5DMVJV4H6JA/bundle.json","state_url":"https://pith.science/pith/PFWF5GBKVQLWO6W5DMVJV4H6JA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PFWF5GBKVQLWO6W5DMVJV4H6JA/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":"2026-06-07T20:47:11Z","links":{"resolver":"https://pith.science/pith/PFWF5GBKVQLWO6W5DMVJV4H6JA","bundle":"https://pith.science/pith/PFWF5GBKVQLWO6W5DMVJV4H6JA/bundle.json","state":"https://pith.science/pith/PFWF5GBKVQLWO6W5DMVJV4H6JA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PFWF5GBKVQLWO6W5DMVJV4H6JA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:PFWF5GBKVQLWO6W5DMVJV4H6JA","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":"c14c560dcc66fd7740a86c8951080d4364e1938c2c3c15d43f18ad5914e4d02d","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-10T16:13:54Z","title_canon_sha256":"a7a200e0730358abc6972dcc0106d0aaa686bcd73fa8baf1ffa48bb53a9a50f5"},"schema_version":"1.0","source":{"id":"1208.2210","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.2210","created_at":"2026-05-18T03:48:59Z"},{"alias_kind":"arxiv_version","alias_value":"1208.2210v1","created_at":"2026-05-18T03:48:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.2210","created_at":"2026-05-18T03:48:59Z"},{"alias_kind":"pith_short_12","alias_value":"PFWF5GBKVQLW","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"PFWF5GBKVQLWO6W5","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"PFWF5GBK","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:2c69bcea0b1460329e67318da169aee06b8143342fb1ed8deb26833fdc363f38","target":"graph","created_at":"2026-05-18T03:48:59Z","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":"Andrews, Lewis and Lovejoy introduced the partition function PD(n) as the number of partitions of $n$ with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that PD(3n+2) is divisible by 3. We obtain a Ramanujan type identity for the generating function of PD(3n+2) which implies the congruence of Andrews, Lewis and Lovejoy. For PD(3n), Andrews, Lewis and Lovejoy showed that the generating function can be expressed as an infinite product of powers of $(1-q^{2n+1})$ times a function $F(q^2)$. We find an explicit formula for $F(q^2)$, wh","authors_text":"Erin Y. Y. Shen, Hai-Tao Jin, Kathy Q. Ji, William Y. C. Chen","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-10T16:13:54Z","title":"On the Number of Partitions with Designated Summands"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.2210","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":"sha256:149e2f58feac09e5b5d4686dc8907302f75821aa5a63766231ede5460eb064ce","target":"record","created_at":"2026-05-18T03:48:59Z","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":"c14c560dcc66fd7740a86c8951080d4364e1938c2c3c15d43f18ad5914e4d02d","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-10T16:13:54Z","title_canon_sha256":"a7a200e0730358abc6972dcc0106d0aaa686bcd73fa8baf1ffa48bb53a9a50f5"},"schema_version":"1.0","source":{"id":"1208.2210","kind":"arxiv","version":1}},"canonical_sha256":"796c5e982aac17677add1b2a9af0fe4823607b2e573630e9bfbc4436f5e84615","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"796c5e982aac17677add1b2a9af0fe4823607b2e573630e9bfbc4436f5e84615","first_computed_at":"2026-05-18T03:48:59.872419Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:48:59.872419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kOR7+kdDtU4oO6hRfL3BQQj/sEdSwAztRbUCfWRf++ODXJvK/4z+qKtgBkJpL9LMZNVc7i8OztNftAykpTR6Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:48:59.873313Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.2210","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:149e2f58feac09e5b5d4686dc8907302f75821aa5a63766231ede5460eb064ce","sha256:2c69bcea0b1460329e67318da169aee06b8143342fb1ed8deb26833fdc363f38"],"state_sha256":"ffe05180257c2713eea07952285097cbdba8966be84697519821323688e24253"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0kI/FOEncWsbz3njcueYPTBUcFINqAQUoKBVM+51YVjkIOu6zaSdSL4IHFAAkZXV47p1KIqT6SWqkLBD1hRtBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T20:47:11.995960Z","bundle_sha256":"c3a849a293de5c05f3370159024930a07b94b6f0290a2c1b8f1e4bb9da203a8d"}}