{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:ZKWYBVST65KM6TGXLKQXVWTOCZ","short_pith_number":"pith:ZKWYBVST","canonical_record":{"source":{"id":"1309.4640","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-18T13:30:11Z","cross_cats_sorted":[],"title_canon_sha256":"955b6cbd095ad81f2baefa7b60d4f46a9a8e3e73daa6df4734cc5a75b734d4d8","abstract_canon_sha256":"631c7c61656ff1e93ebd7275cd2ee4b5b01ecd57b1fa6489afe8686c84f43dd9"},"schema_version":"1.0"},"canonical_sha256":"caad80d653f754cf4cd75aa17ada6e164cda9063fcc5c44972568351bfc49ca9","source":{"kind":"arxiv","id":"1309.4640","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.4640","created_at":"2026-05-18T03:12:58Z"},{"alias_kind":"arxiv_version","alias_value":"1309.4640v1","created_at":"2026-05-18T03:12:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.4640","created_at":"2026-05-18T03:12:58Z"},{"alias_kind":"pith_short_12","alias_value":"ZKWYBVST65KM","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"ZKWYBVST65KM6TGX","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"ZKWYBVST","created_at":"2026-05-18T12:28:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:ZKWYBVST65KM6TGXLKQXVWTOCZ","target":"record","payload":{"canonical_record":{"source":{"id":"1309.4640","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-18T13:30:11Z","cross_cats_sorted":[],"title_canon_sha256":"955b6cbd095ad81f2baefa7b60d4f46a9a8e3e73daa6df4734cc5a75b734d4d8","abstract_canon_sha256":"631c7c61656ff1e93ebd7275cd2ee4b5b01ecd57b1fa6489afe8686c84f43dd9"},"schema_version":"1.0"},"canonical_sha256":"caad80d653f754cf4cd75aa17ada6e164cda9063fcc5c44972568351bfc49ca9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:58.736697Z","signature_b64":"El5Q+CWobYUnMFcJnCYBbODbnPNQ9CMTOBI/i07zdMn945JwsWqIAcG8Q4jmx8d9k+kZscJb1hgZ/C829xxGBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"caad80d653f754cf4cd75aa17ada6e164cda9063fcc5c44972568351bfc49ca9","last_reissued_at":"2026-05-18T03:12:58.736024Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:58.736024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.4640","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":"2026-05-18T03:12:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OPm/eZI9x1HaLqZYDlUK18k+FNjdJz8f3x97uGA5ePQFfjxtXoGoOAuQ+PFtbVEJMoOluSp5ynfTSbCVzkM9Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T13:59:43.697404Z"},"content_sha256":"4608733c568a1cc4fc70cbaf7881060579ddf18393a560d545eeb66303ed18db","schema_version":"1.0","event_id":"sha256:4608733c568a1cc4fc70cbaf7881060579ddf18393a560d545eeb66303ed18db"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:ZKWYBVST65KM6TGXLKQXVWTOCZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Proof of two conjectures of Ciucu and Krattenthaler on the enumeration of lozenge tilings of hexagons with cut off corners","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ilse Fischer, Mihai Ciucu","submitted_at":"2013-09-18T13:30:11Z","abstract_excerpt":"In their 2002 paper, Ciucu and Krattenthaler proved several product formulas for the number of lozenge tilings of various regions obtained from a centrally symmetric hexagon on the triangular lattice by removing maximal staircase regions from two non-adjacent corners. For the case when the staircases are removed from adjacent corners of the hexagon, they presented two conjectural formulas, whose proofs, as they remarked, seemed at the time \"a formidable task\". In this paper we prove those two conjectures. Our proofs proceed by first generalizing the conjectures, and then proving them by induct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4640","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":"2026-05-18T03:12:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9Do/nLuAb96oRze3xRxXHIuyAFOSkKDN/a5N80dh8pCvuvC738WM9cQt1DI9+tCyB3J4DDNUQsplaacH+9DgCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T13:59:43.697756Z"},"content_sha256":"50063b963475f9e3ab0161591f1b506d1684b56b505d0161d10bea31120b9b73","schema_version":"1.0","event_id":"sha256:50063b963475f9e3ab0161591f1b506d1684b56b505d0161d10bea31120b9b73"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZKWYBVST65KM6TGXLKQXVWTOCZ/bundle.json","state_url":"https://pith.science/pith/ZKWYBVST65KM6TGXLKQXVWTOCZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZKWYBVST65KM6TGXLKQXVWTOCZ/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":"2026-06-22T13:59:43Z","links":{"resolver":"https://pith.science/pith/ZKWYBVST65KM6TGXLKQXVWTOCZ","bundle":"https://pith.science/pith/ZKWYBVST65KM6TGXLKQXVWTOCZ/bundle.json","state":"https://pith.science/pith/ZKWYBVST65KM6TGXLKQXVWTOCZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZKWYBVST65KM6TGXLKQXVWTOCZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ZKWYBVST65KM6TGXLKQXVWTOCZ","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":"631c7c61656ff1e93ebd7275cd2ee4b5b01ecd57b1fa6489afe8686c84f43dd9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-18T13:30:11Z","title_canon_sha256":"955b6cbd095ad81f2baefa7b60d4f46a9a8e3e73daa6df4734cc5a75b734d4d8"},"schema_version":"1.0","source":{"id":"1309.4640","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.4640","created_at":"2026-05-18T03:12:58Z"},{"alias_kind":"arxiv_version","alias_value":"1309.4640v1","created_at":"2026-05-18T03:12:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.4640","created_at":"2026-05-18T03:12:58Z"},{"alias_kind":"pith_short_12","alias_value":"ZKWYBVST65KM","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"ZKWYBVST65KM6TGX","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"ZKWYBVST","created_at":"2026-05-18T12:28:09Z"}],"graph_snapshots":[{"event_id":"sha256:50063b963475f9e3ab0161591f1b506d1684b56b505d0161d10bea31120b9b73","target":"graph","created_at":"2026-05-18T03:12:58Z","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":"In their 2002 paper, Ciucu and Krattenthaler proved several product formulas for the number of lozenge tilings of various regions obtained from a centrally symmetric hexagon on the triangular lattice by removing maximal staircase regions from two non-adjacent corners. For the case when the staircases are removed from adjacent corners of the hexagon, they presented two conjectural formulas, whose proofs, as they remarked, seemed at the time \"a formidable task\". In this paper we prove those two conjectures. Our proofs proceed by first generalizing the conjectures, and then proving them by induct","authors_text":"Ilse Fischer, Mihai Ciucu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-18T13:30:11Z","title":"Proof of two conjectures of Ciucu and Krattenthaler on the enumeration of lozenge tilings of hexagons with cut off corners"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4640","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":"sha256:4608733c568a1cc4fc70cbaf7881060579ddf18393a560d545eeb66303ed18db","target":"record","created_at":"2026-05-18T03:12:58Z","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":"631c7c61656ff1e93ebd7275cd2ee4b5b01ecd57b1fa6489afe8686c84f43dd9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-18T13:30:11Z","title_canon_sha256":"955b6cbd095ad81f2baefa7b60d4f46a9a8e3e73daa6df4734cc5a75b734d4d8"},"schema_version":"1.0","source":{"id":"1309.4640","kind":"arxiv","version":1}},"canonical_sha256":"caad80d653f754cf4cd75aa17ada6e164cda9063fcc5c44972568351bfc49ca9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"caad80d653f754cf4cd75aa17ada6e164cda9063fcc5c44972568351bfc49ca9","first_computed_at":"2026-05-18T03:12:58.736024Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:58.736024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"El5Q+CWobYUnMFcJnCYBbODbnPNQ9CMTOBI/i07zdMn945JwsWqIAcG8Q4jmx8d9k+kZscJb1hgZ/C829xxGBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:58.736697Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.4640","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4608733c568a1cc4fc70cbaf7881060579ddf18393a560d545eeb66303ed18db","sha256:50063b963475f9e3ab0161591f1b506d1684b56b505d0161d10bea31120b9b73"],"state_sha256":"3b20b4763b853183021254d919fc24614a603010404683fdcb62bef69da19597"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TiCgTofEz6a5kaVK6w2Q8OmJ/0ptp7NaNY71hRTahB1/BlpEzQc3k8z3byWRODBN2IBvu7XzZmfzb6Lq2JApDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T13:59:43.699928Z","bundle_sha256":"0038eccaf26b0ef0494f846c907e37551ce8b3c0841ff7308fb3dc866e8259d4"}}