{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:4NAXKYIH3CBIICVDDTC43BJ7HT","short_pith_number":"pith:4NAXKYIH","schema_version":"1.0","canonical_sha256":"e341756107d882840aa31cc5cd853f3cccbfe5fd39f2fc48aaa62b5e7d5d489d","source":{"kind":"arxiv","id":"1603.03259","version":4},"attestation_state":"computed","paper":{"title":"Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.RA","nlin.SI"],"primary_cat":"math.CO","authors_text":"Adam Doliwa","submitted_at":"2016-03-10T13:37:36Z","abstract_excerpt":"We introduce a coloured generalization $\\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables. We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various"},"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":"1603.03259","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-03-10T13:37:36Z","cross_cats_sorted":["math-ph","math.MP","math.RA","nlin.SI"],"title_canon_sha256":"15bf5f19d503f8af7b2dd74101afd457b6a4e94bdfe363c24f115e95e8f0f310","abstract_canon_sha256":"f8682c9d31b318ffce19b11d8d20413922021be56652c2ec7cb46269ff45e728"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:54:11.510435Z","signature_b64":"xP4w8n+ZuCKWwxANNRjiacjEeO71V31DjHI3Cwuf3n0Vwtv7aQ4nhrzt5AqbG0dyoWLfH6Xx3jEXlI87CY4wAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e341756107d882840aa31cc5cd853f3cccbfe5fd39f2fc48aaa62b5e7d5d489d","last_reissued_at":"2026-07-05T02:54:11.510051Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:54:11.510051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.RA","nlin.SI"],"primary_cat":"math.CO","authors_text":"Adam Doliwa","submitted_at":"2016-03-10T13:37:36Z","abstract_excerpt":"We introduce a coloured generalization $\\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables. We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.03259","kind":"arxiv","version":4},"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/1603.03259/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":"1603.03259","created_at":"2026-07-05T02:54:11.510107+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.03259v4","created_at":"2026-07-05T02:54:11.510107+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.03259","created_at":"2026-07-05T02:54:11.510107+00:00"},{"alias_kind":"pith_short_12","alias_value":"4NAXKYIH3CBI","created_at":"2026-07-05T02:54:11.510107+00:00"},{"alias_kind":"pith_short_16","alias_value":"4NAXKYIH3CBIICVD","created_at":"2026-07-05T02:54:11.510107+00:00"},{"alias_kind":"pith_short_8","alias_value":"4NAXKYIH","created_at":"2026-07-05T02:54:11.510107+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"1907.09975","citing_title":"Hopf algebra structure of symmetric and quasisymmetric functions in superspace","ref_index":3,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4NAXKYIH3CBIICVDDTC43BJ7HT","json":"https://pith.science/pith/4NAXKYIH3CBIICVDDTC43BJ7HT.json","graph_json":"https://pith.science/api/pith-number/4NAXKYIH3CBIICVDDTC43BJ7HT/graph.json","events_json":"https://pith.science/api/pith-number/4NAXKYIH3CBIICVDDTC43BJ7HT/events.json","paper":"https://pith.science/paper/4NAXKYIH"},"agent_actions":{"view_html":"https://pith.science/pith/4NAXKYIH3CBIICVDDTC43BJ7HT","download_json":"https://pith.science/pith/4NAXKYIH3CBIICVDDTC43BJ7HT.json","view_paper":"https://pith.science/paper/4NAXKYIH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.03259&json=true","fetch_graph":"https://pith.science/api/pith-number/4NAXKYIH3CBIICVDDTC43BJ7HT/graph.json","fetch_events":"https://pith.science/api/pith-number/4NAXKYIH3CBIICVDDTC43BJ7HT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4NAXKYIH3CBIICVDDTC43BJ7HT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4NAXKYIH3CBIICVDDTC43BJ7HT/action/storage_attestation","attest_author":"https://pith.science/pith/4NAXKYIH3CBIICVDDTC43BJ7HT/action/author_attestation","sign_citation":"https://pith.science/pith/4NAXKYIH3CBIICVDDTC43BJ7HT/action/citation_signature","submit_replication":"https://pith.science/pith/4NAXKYIH3CBIICVDDTC43BJ7HT/action/replication_record"}},"created_at":"2026-07-05T02:54:11.510107+00:00","updated_at":"2026-07-05T02:54:11.510107+00:00"}