{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:V5SLURL3UBVVH63EZVULKCZ3FO","short_pith_number":"pith:V5SLURL3","schema_version":"1.0","canonical_sha256":"af64ba457ba06b53fb64cd68b50b3b2ba384b4d64706a990621a4a043e78d622","source":{"kind":"arxiv","id":"2003.03896","version":1},"attestation_state":"computed","paper":{"title":"Chain Decompositions of $q,t$-Catalan Numbers via Local Chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kyungyong Lee, Li Li, Nicholas A. Loehr, Seongjune Han","submitted_at":"2020-03-09T02:43:19Z","abstract_excerpt":"The $q,t$-Catalan number $\\mathrm{Cat}_n(q,t)$ enumerates integer partitions contained in an $n\\times n$ triangle by their dinv and external area statistics. The paper [LLL18 (Lee, Li, Loehr, SIAM J. Discrete Math. 32(2018))] proposed a new approach to understanding the symmetry property $\\mathrm{Cat}_n(q,t)=\\mathrm{Cat}_n(t,q)$ based on decomposing the set of all integer partitions into infinite chains. Each such global chain $\\mathcal{C}_{\\mu}$ has an opposite chain $\\mathcal{C}_{\\mu^*}$; these combine to give a new small slice of $\\mathrm{Cat}_n(q,t)$ that is symmetric in $q$ and $t$. Here "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2003.03896","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-03-09T02:43:19Z","cross_cats_sorted":[],"title_canon_sha256":"8c343433e6399a75fa8c6cd2fd23bd857ad267734ad054aefdb7466f425041b8","abstract_canon_sha256":"602a15f75c825cb518018c824791ed3652d321e42cae24dacb7ba2bca6b3d195"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:46:37.602941Z","signature_b64":"ybWANLw46S28I2dPFd7HyelzxHaRgG9ulYiYkhwIM/PmugvMLZkeosOGdFkKkgu7j9WDQoV0zkz8qd+bsKamBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af64ba457ba06b53fb64cd68b50b3b2ba384b4d64706a990621a4a043e78d622","last_reissued_at":"2026-07-05T00:46:37.602467Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:46:37.602467Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Chain Decompositions of $q,t$-Catalan Numbers via Local Chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kyungyong Lee, Li Li, Nicholas A. Loehr, Seongjune Han","submitted_at":"2020-03-09T02:43:19Z","abstract_excerpt":"The $q,t$-Catalan number $\\mathrm{Cat}_n(q,t)$ enumerates integer partitions contained in an $n\\times n$ triangle by their dinv and external area statistics. The paper [LLL18 (Lee, Li, Loehr, SIAM J. Discrete Math. 32(2018))] proposed a new approach to understanding the symmetry property $\\mathrm{Cat}_n(q,t)=\\mathrm{Cat}_n(t,q)$ based on decomposing the set of all integer partitions into infinite chains. Each such global chain $\\mathcal{C}_{\\mu}$ has an opposite chain $\\mathcal{C}_{\\mu^*}$; these combine to give a new small slice of $\\mathrm{Cat}_n(q,t)$ that is symmetric in $q$ and $t$. Here "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2003.03896","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2003.03896/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2003.03896","created_at":"2026-07-05T00:46:37.602524+00:00"},{"alias_kind":"arxiv_version","alias_value":"2003.03896v1","created_at":"2026-07-05T00:46:37.602524+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2003.03896","created_at":"2026-07-05T00:46:37.602524+00:00"},{"alias_kind":"pith_short_12","alias_value":"V5SLURL3UBVV","created_at":"2026-07-05T00:46:37.602524+00:00"},{"alias_kind":"pith_short_16","alias_value":"V5SLURL3UBVVH63E","created_at":"2026-07-05T00:46:37.602524+00:00"},{"alias_kind":"pith_short_8","alias_value":"V5SLURL3","created_at":"2026-07-05T00:46:37.602524+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/V5SLURL3UBVVH63EZVULKCZ3FO","json":"https://pith.science/pith/V5SLURL3UBVVH63EZVULKCZ3FO.json","graph_json":"https://pith.science/api/pith-number/V5SLURL3UBVVH63EZVULKCZ3FO/graph.json","events_json":"https://pith.science/api/pith-number/V5SLURL3UBVVH63EZVULKCZ3FO/events.json","paper":"https://pith.science/paper/V5SLURL3"},"agent_actions":{"view_html":"https://pith.science/pith/V5SLURL3UBVVH63EZVULKCZ3FO","download_json":"https://pith.science/pith/V5SLURL3UBVVH63EZVULKCZ3FO.json","view_paper":"https://pith.science/paper/V5SLURL3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2003.03896&json=true","fetch_graph":"https://pith.science/api/pith-number/V5SLURL3UBVVH63EZVULKCZ3FO/graph.json","fetch_events":"https://pith.science/api/pith-number/V5SLURL3UBVVH63EZVULKCZ3FO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V5SLURL3UBVVH63EZVULKCZ3FO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V5SLURL3UBVVH63EZVULKCZ3FO/action/storage_attestation","attest_author":"https://pith.science/pith/V5SLURL3UBVVH63EZVULKCZ3FO/action/author_attestation","sign_citation":"https://pith.science/pith/V5SLURL3UBVVH63EZVULKCZ3FO/action/citation_signature","submit_replication":"https://pith.science/pith/V5SLURL3UBVVH63EZVULKCZ3FO/action/replication_record"}},"created_at":"2026-07-05T00:46:37.602524+00:00","updated_at":"2026-07-05T00:46:37.602524+00:00"}