{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:RUUTGZUOHMQBPEH7C3APFMC7QD","short_pith_number":"pith:RUUTGZUO","canonical_record":{"source":{"id":"1906.04532","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-11T12:37:23Z","cross_cats_sorted":[],"title_canon_sha256":"be3ecd40c639b7f058d4841cf7e12c1557a4a914e6ca7a3da45c271f9b3ac0af","abstract_canon_sha256":"316248d69a34b7377f16e974b8d279f6fcf691f365b67dca12a84b1713af7bba"},"schema_version":"1.0"},"canonical_sha256":"8d2933668e3b201790ff16c0f2b05f80ff996f75a5da71b68801ea4d6084bdad","source":{"kind":"arxiv","id":"1906.04532","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.04532","created_at":"2026-05-17T23:43:39Z"},{"alias_kind":"arxiv_version","alias_value":"1906.04532v1","created_at":"2026-05-17T23:43:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.04532","created_at":"2026-05-17T23:43:39Z"},{"alias_kind":"pith_short_12","alias_value":"RUUTGZUOHMQB","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"RUUTGZUOHMQBPEH7","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"RUUTGZUO","created_at":"2026-05-18T12:33:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:RUUTGZUOHMQBPEH7C3APFMC7QD","target":"record","payload":{"canonical_record":{"source":{"id":"1906.04532","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-11T12:37:23Z","cross_cats_sorted":[],"title_canon_sha256":"be3ecd40c639b7f058d4841cf7e12c1557a4a914e6ca7a3da45c271f9b3ac0af","abstract_canon_sha256":"316248d69a34b7377f16e974b8d279f6fcf691f365b67dca12a84b1713af7bba"},"schema_version":"1.0"},"canonical_sha256":"8d2933668e3b201790ff16c0f2b05f80ff996f75a5da71b68801ea4d6084bdad","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:39.017743Z","signature_b64":"YPwgUeK0lPX45koQAD/0F4j7fe6Q5O2wBL2N0yZ+PAirkUd1QugvBLW6gVp+nFywZrP7k5MzPuYUk82TqkZjBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8d2933668e3b201790ff16c0f2b05f80ff996f75a5da71b68801ea4d6084bdad","last_reissued_at":"2026-05-17T23:43:39.017172Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:39.017172Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1906.04532","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":"2026-05-17T23:43:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m7RPYmQgKog5u8wPINjrg8Z69VD2bJSR4YjH5MsnBCiEGZj2EuoRlbXQ/VkY9IQ09ZlGEctgIWCELLlRDK42AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T02:26:21.237347Z"},"content_sha256":"e69980e2eb8320b58b558bd97ab49e2ad9708bf46a63d22233d9534da3194301","schema_version":"1.0","event_id":"sha256:e69980e2eb8320b58b558bd97ab49e2ad9708bf46a63d22233d9534da3194301"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:RUUTGZUOHMQBPEH7C3APFMC7QD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Three product formulas for ratios of tiling counts of hexagons with collinear holes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Seok Hyun Byun","submitted_at":"2019-06-11T12:37:23Z","abstract_excerpt":"Rosengren found an explicit formula for a certain weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole. He pointed out that a certain ratio corresponding to two such regions has a nice product formula. In this paper, we generalize this to hexagons with arbitrary collinear holes. It turns out that, by using same approach, we can also generalize Ciucu's work on the number and the number of centrally symmetric tilings of a hexagon with a fern removed from its center. This proves a recent conjecture of Ciucu."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.04532","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":"2026-05-17T23:43:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BUXBXAY0hbQpMjsLH9hT0MjEFusARCBiDdV7inuOWAwZUiXCBwrfmWoYYbArsiWR/p3QNbfhW4A8dRTAh81ZAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T02:26:21.237714Z"},"content_sha256":"5362aef9053da409788df7ed2fc43c6903645062f54f022e4c16d3e0dcbcc9c4","schema_version":"1.0","event_id":"sha256:5362aef9053da409788df7ed2fc43c6903645062f54f022e4c16d3e0dcbcc9c4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RUUTGZUOHMQBPEH7C3APFMC7QD/bundle.json","state_url":"https://pith.science/pith/RUUTGZUOHMQBPEH7C3APFMC7QD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RUUTGZUOHMQBPEH7C3APFMC7QD/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":"2026-05-30T02:26:21Z","links":{"resolver":"https://pith.science/pith/RUUTGZUOHMQBPEH7C3APFMC7QD","bundle":"https://pith.science/pith/RUUTGZUOHMQBPEH7C3APFMC7QD/bundle.json","state":"https://pith.science/pith/RUUTGZUOHMQBPEH7C3APFMC7QD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RUUTGZUOHMQBPEH7C3APFMC7QD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:RUUTGZUOHMQBPEH7C3APFMC7QD","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":"316248d69a34b7377f16e974b8d279f6fcf691f365b67dca12a84b1713af7bba","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-11T12:37:23Z","title_canon_sha256":"be3ecd40c639b7f058d4841cf7e12c1557a4a914e6ca7a3da45c271f9b3ac0af"},"schema_version":"1.0","source":{"id":"1906.04532","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.04532","created_at":"2026-05-17T23:43:39Z"},{"alias_kind":"arxiv_version","alias_value":"1906.04532v1","created_at":"2026-05-17T23:43:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.04532","created_at":"2026-05-17T23:43:39Z"},{"alias_kind":"pith_short_12","alias_value":"RUUTGZUOHMQB","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"RUUTGZUOHMQBPEH7","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"RUUTGZUO","created_at":"2026-05-18T12:33:27Z"}],"graph_snapshots":[{"event_id":"sha256:5362aef9053da409788df7ed2fc43c6903645062f54f022e4c16d3e0dcbcc9c4","target":"graph","created_at":"2026-05-17T23:43:39Z","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":"Rosengren found an explicit formula for a certain weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole. He pointed out that a certain ratio corresponding to two such regions has a nice product formula. In this paper, we generalize this to hexagons with arbitrary collinear holes. It turns out that, by using same approach, we can also generalize Ciucu's work on the number and the number of centrally symmetric tilings of a hexagon with a fern removed from its center. This proves a recent conjecture of Ciucu.","authors_text":"Seok Hyun Byun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-11T12:37:23Z","title":"Three product formulas for ratios of tiling counts of hexagons with collinear holes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.04532","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:e69980e2eb8320b58b558bd97ab49e2ad9708bf46a63d22233d9534da3194301","target":"record","created_at":"2026-05-17T23:43:39Z","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":"316248d69a34b7377f16e974b8d279f6fcf691f365b67dca12a84b1713af7bba","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-11T12:37:23Z","title_canon_sha256":"be3ecd40c639b7f058d4841cf7e12c1557a4a914e6ca7a3da45c271f9b3ac0af"},"schema_version":"1.0","source":{"id":"1906.04532","kind":"arxiv","version":1}},"canonical_sha256":"8d2933668e3b201790ff16c0f2b05f80ff996f75a5da71b68801ea4d6084bdad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8d2933668e3b201790ff16c0f2b05f80ff996f75a5da71b68801ea4d6084bdad","first_computed_at":"2026-05-17T23:43:39.017172Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:43:39.017172Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YPwgUeK0lPX45koQAD/0F4j7fe6Q5O2wBL2N0yZ+PAirkUd1QugvBLW6gVp+nFywZrP7k5MzPuYUk82TqkZjBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:43:39.017743Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.04532","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e69980e2eb8320b58b558bd97ab49e2ad9708bf46a63d22233d9534da3194301","sha256:5362aef9053da409788df7ed2fc43c6903645062f54f022e4c16d3e0dcbcc9c4"],"state_sha256":"a77f4cdd72c44339d3d70f1fc60be5c9ab44aaf604a59d49fa71872662a4df04"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yhq/SOYNDec3L+XZ8sx5ET9U94nWmJOBqUpgwm7k6rookpK2qruaUC63Iw6v9eJ1xcECCmH/Tq/m5UCY5y7JCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T02:26:21.239792Z","bundle_sha256":"86466eca4a53ca83a10beb81e75c785246bcd75397c1c2a0d193ebc37ec67ffd"}}