{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:GSDRZH4JBLHSK2KV2TZIMLXZO4","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":"e538ed7949fe3850a74ea1979b8fbf1aca85e48357f9d964bfa91fa3cf1ea980","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-04-18T02:50:26Z","title_canon_sha256":"7d93d1870140f29d8a94a825e30b938b414c3866a99ce0355cfcf9a4116066d9"},"schema_version":"1.0","source":{"id":"0804.2930","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0804.2930","created_at":"2026-07-04T15:13:10Z"},{"alias_kind":"arxiv_version","alias_value":"0804.2930v2","created_at":"2026-07-04T15:13:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0804.2930","created_at":"2026-07-04T15:13:10Z"},{"alias_kind":"pith_short_12","alias_value":"GSDRZH4JBLHS","created_at":"2026-07-04T15:13:10Z"},{"alias_kind":"pith_short_16","alias_value":"GSDRZH4JBLHSK2KV","created_at":"2026-07-04T15:13:10Z"},{"alias_kind":"pith_short_8","alias_value":"GSDRZH4J","created_at":"2026-07-04T15:13:10Z"}],"graph_snapshots":[{"event_id":"sha256:d1506e8d3984f16af82ec24d2ea83c5f20a5c4c15c13c8936f02708cc819f665","target":"graph","created_at":"2026-07-04T15:13:10Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/0804.2930/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length $2n$ and noncrossing partitions of $[2n+1]$ with $n+1$ blocks. In terms of the number of up steps at odd positions, we find a characterization of Dyck paths constructed from pairs of noncrossing free Dyck paths by using the Labelle merging algorithm.","authors_text":"Ellen X.Y. Qu, Richard P. Stanley, Sabrina X.M. Pang, William Y.C. Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-04-18T02:50:26Z","title":"Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0804.2930","kind":"arxiv","version":2},"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":"sha256:80ea14e963e20c9cb70aad8b965101400aa5971760657a7bdf3f568a4675b582","target":"record","created_at":"2026-07-04T15:13:10Z","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":"e538ed7949fe3850a74ea1979b8fbf1aca85e48357f9d964bfa91fa3cf1ea980","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-04-18T02:50:26Z","title_canon_sha256":"7d93d1870140f29d8a94a825e30b938b414c3866a99ce0355cfcf9a4116066d9"},"schema_version":"1.0","source":{"id":"0804.2930","kind":"arxiv","version":2}},"canonical_sha256":"34871c9f890acf256955d4f2862ef97733b49bd8cac58ace903ffca7582bdcc3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"34871c9f890acf256955d4f2862ef97733b49bd8cac58ace903ffca7582bdcc3","first_computed_at":"2026-07-04T15:13:10.255765Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:13:10.255765Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pd/OEKKitYdYoPEBElLvT1syX1X53foMdFBrjxMAi1WmmRLm92yJuFs+MTLRNU9SlreCOV6h3Iw5TmIwbjwzDA==","signature_status":"signed_v1","signed_at":"2026-07-04T15:13:10.256204Z","signed_message":"canonical_sha256_bytes"},"source_id":"0804.2930","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:80ea14e963e20c9cb70aad8b965101400aa5971760657a7bdf3f568a4675b582","sha256:d1506e8d3984f16af82ec24d2ea83c5f20a5c4c15c13c8936f02708cc819f665"],"state_sha256":"9e5d97af52c1aa1efb4df38d8e48d3caea956e99e27f9bd014cbd048e48568b1"}