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They can be described in terms of Legendre's beta function evaluated at half-integers, or in terms of bivariate Catalan numbers:\n $$ C(m,n)=\\frac{(2m)!(2n)!}{m!(m+n)!n!}. $$\n Properties of characters and of quasi-symmetric functions are then used to derive several interesting identities amo"},"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":"math/0408053","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2004-08-04T16:54:09Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"816ad83f94de29af2465292cd5cdeab39f49582d298902030cf8b680976b2e2d","abstract_canon_sha256":"03629767b63b5e334ff34a4906b6ccecfece5c1dbf5e97a7d50dd0d3fa6e594f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:26.141106Z","signature_b64":"9G+owNjsp3ZvnyQuNwdrmwMP10PrEBHX1mLlzYF81h2S/B19QJoGTUm9JxNjCrTJ0hMiROKZ1z03dhfeHVdDCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5215eaaaa20afc7596548387dc709750b902bc39e76643fa025b9629bac96669","last_reissued_at":"2026-05-18T01:05:26.140499Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:26.140499Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Canonical characters on quasi-symmetric functions and bivariate Catalan numbers","license":"","headline":"","cross_cats":["math.QA"],"primary_cat":"math.CO","authors_text":"Marcelo Aguiar, Samuel K. 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They can be described in terms of Legendre's beta function evaluated at half-integers, or in terms of bivariate Catalan numbers:\n $$ C(m,n)=\\frac{(2m)!(2n)!}{m!(m+n)!n!}. $$\n Properties of characters and of quasi-symmetric functions are then used to derive several interesting identities amo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0408053","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0408053","created_at":"2026-05-18T01:05:26.140583+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0408053v1","created_at":"2026-05-18T01:05:26.140583+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0408053","created_at":"2026-05-18T01:05:26.140583+00:00"},{"alias_kind":"pith_short_12","alias_value":"KIK6VKVCBL6H","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_16","alias_value":"KIK6VKVCBL6HLFSU","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_8","alias_value":"KIK6VKVC","created_at":"2026-05-18T12:25:52.687210+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/KIK6VKVCBL6HLFSUQOD5Y4EXKC","json":"https://pith.science/pith/KIK6VKVCBL6HLFSUQOD5Y4EXKC.json","graph_json":"https://pith.science/api/pith-number/KIK6VKVCBL6HLFSUQOD5Y4EXKC/graph.json","events_json":"https://pith.science/api/pith-number/KIK6VKVCBL6HLFSUQOD5Y4EXKC/events.json","paper":"https://pith.science/paper/KIK6VKVC"},"agent_actions":{"view_html":"https://pith.science/pith/KIK6VKVCBL6HLFSUQOD5Y4EXKC","download_json":"https://pith.science/pith/KIK6VKVCBL6HLFSUQOD5Y4EXKC.json","view_paper":"https://pith.science/paper/KIK6VKVC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0408053&json=true","fetch_graph":"https://pith.science/api/pith-number/KIK6VKVCBL6HLFSUQOD5Y4EXKC/graph.json","fetch_events":"https://pith.science/api/pith-number/KIK6VKVCBL6HLFSUQOD5Y4EXKC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KIK6VKVCBL6HLFSUQOD5Y4EXKC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KIK6VKVCBL6HLFSUQOD5Y4EXKC/action/storage_attestation","attest_author":"https://pith.science/pith/KIK6VKVCBL6HLFSUQOD5Y4EXKC/action/author_attestation","sign_citation":"https://pith.science/pith/KIK6VKVCBL6HLFSUQOD5Y4EXKC/action/citation_signature","submit_replication":"https://pith.science/pith/KIK6VKVCBL6HLFSUQOD5Y4EXKC/action/replication_record"}},"created_at":"2026-05-18T01:05:26.140583+00:00","updated_at":"2026-05-18T01:05:26.140583+00:00"}