{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:VKAMNMLYGRPRL2KSMQJN4SLROE","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":"e2d4d2a798bfe46ca4c674a4a13ad98a72a545c87524b9d3c8109e1b569e5f3b","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-29T14:13:33Z","title_canon_sha256":"52c3958ce5a2de508d83c620619eca1cd0ab22cb0dc2340d769dffad56378607"},"schema_version":"1.0","source":{"id":"1801.09516","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.09516","created_at":"2026-05-18T00:24:57Z"},{"alias_kind":"arxiv_version","alias_value":"1801.09516v1","created_at":"2026-05-18T00:24:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.09516","created_at":"2026-05-18T00:24:57Z"},{"alias_kind":"pith_short_12","alias_value":"VKAMNMLYGRPR","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VKAMNMLYGRPRL2KS","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VKAMNMLY","created_at":"2026-05-18T12:32:59Z"}],"graph_snapshots":[{"event_id":"sha256:8b18584b8355e67b2f50874244c088dd23a4ca1a20f0667786b368d09766b314","target":"graph","created_at":"2026-05-18T00:24:57Z","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":"A bound resembling Pascal's identity is presented for binary necklaces with fixed density using Lyndon words with fixed density. The result is generalized to k-ary necklaces and Lyndon words with fixed content. The bound arises in the study of Nichols algebras of diagonal type.","authors_text":"I. Heckenberger, J. Sawada","cross_cats":["math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-29T14:13:33Z","title":"A Pascal-like Bound for the Number of Necklaces with Fixed Density"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.09516","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":"sha256:32bd564b5fa309129fcdd6508ca48f1154697a19c6df652d3f14c0b355a8344c","target":"record","created_at":"2026-05-18T00:24:57Z","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":"e2d4d2a798bfe46ca4c674a4a13ad98a72a545c87524b9d3c8109e1b569e5f3b","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-29T14:13:33Z","title_canon_sha256":"52c3958ce5a2de508d83c620619eca1cd0ab22cb0dc2340d769dffad56378607"},"schema_version":"1.0","source":{"id":"1801.09516","kind":"arxiv","version":1}},"canonical_sha256":"aa80c6b178345f15e9526412de49717118a35b5a01d7b7cdc1a6ec91197a1210","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aa80c6b178345f15e9526412de49717118a35b5a01d7b7cdc1a6ec91197a1210","first_computed_at":"2026-05-18T00:24:57.191457Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:24:57.191457Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"M+fSuTzhkcL2Bmi3MAZhxJnCHdiAuhv3FgCaeY7lW7UlI9YO9e3eCBPzv8IzwkFDFFnDdp+x266uPqrSW04BDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:24:57.192051Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.09516","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:32bd564b5fa309129fcdd6508ca48f1154697a19c6df652d3f14c0b355a8344c","sha256:8b18584b8355e67b2f50874244c088dd23a4ca1a20f0667786b368d09766b314"],"state_sha256":"b2f7e89c01c587d273e0951b8cdb18b8582d7247a89bd07a90b3239be2a20b27"}