{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:2WL6FSOX3WY3VC6F5RNENKPMU3","short_pith_number":"pith:2WL6FSOX","schema_version":"1.0","canonical_sha256":"d597e2c9d7ddb1ba8bc5ec5a46a9eca6c536d030b888dbd77028ca410752bff9","source":{"kind":"arxiv","id":"1512.01637","version":1},"attestation_state":"computed","paper":{"title":"The mechanics of shuffle products and their siblings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christophe Tollu (LIPN), G\\'erard H. E. Duchamp (LIPN), Jean-Yves Enjalbert (LIPN), Vincel Hoang (LIPN)","submitted_at":"2015-12-05T08:15:07Z","abstract_excerpt":"We carry on the investigation initiated in [15] : we describe new shuffle products coming from some special functions and group them, along with other products encountered in the literature, in a class of products, which we name $\\varphi$-shuffle products. Our paper is dedicated to a study of the latter class, from a combinatorial standpoint. We consider first how to extend Radford's theorem to the products in that class, then how to construct their bi-algebras. As some conditions are necessary do carry that out, we study them closely and simplify them so that they can be seen directly from th"},"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":"1512.01637","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-05T08:15:07Z","cross_cats_sorted":[],"title_canon_sha256":"1742a0705620709ca17d460476b8b48da594ce6aff9d95b90f10eaf88bec65f8","abstract_canon_sha256":"c0ff2c0a387dbf579f28594a17374f36a983b50530b746a972a16ebd3edcd5df"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:10.755139Z","signature_b64":"/rvA2fYZ8nVEcD8IxHS43RU2CX7lYN/4NK90kkVLCK70hgoiihb+stqrtPK2WfUsAsx7paZsTMS9HtMxSVBLBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d597e2c9d7ddb1ba8bc5ec5a46a9eca6c536d030b888dbd77028ca410752bff9","last_reissued_at":"2026-05-18T01:25:10.754532Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:10.754532Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The mechanics of shuffle products and their siblings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christophe Tollu (LIPN), G\\'erard H. E. Duchamp (LIPN), Jean-Yves Enjalbert (LIPN), Vincel Hoang (LIPN)","submitted_at":"2015-12-05T08:15:07Z","abstract_excerpt":"We carry on the investigation initiated in [15] : we describe new shuffle products coming from some special functions and group them, along with other products encountered in the literature, in a class of products, which we name $\\varphi$-shuffle products. Our paper is dedicated to a study of the latter class, from a combinatorial standpoint. We consider first how to extend Radford's theorem to the products in that class, then how to construct their bi-algebras. As some conditions are necessary do carry that out, we study them closely and simplify them so that they can be seen directly from th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01637","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":"1512.01637","created_at":"2026-05-18T01:25:10.754628+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.01637v1","created_at":"2026-05-18T01:25:10.754628+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01637","created_at":"2026-05-18T01:25:10.754628+00:00"},{"alias_kind":"pith_short_12","alias_value":"2WL6FSOX3WY3","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_16","alias_value":"2WL6FSOX3WY3VC6F","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_8","alias_value":"2WL6FSOX","created_at":"2026-05-18T12:29:02.477457+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/2WL6FSOX3WY3VC6F5RNENKPMU3","json":"https://pith.science/pith/2WL6FSOX3WY3VC6F5RNENKPMU3.json","graph_json":"https://pith.science/api/pith-number/2WL6FSOX3WY3VC6F5RNENKPMU3/graph.json","events_json":"https://pith.science/api/pith-number/2WL6FSOX3WY3VC6F5RNENKPMU3/events.json","paper":"https://pith.science/paper/2WL6FSOX"},"agent_actions":{"view_html":"https://pith.science/pith/2WL6FSOX3WY3VC6F5RNENKPMU3","download_json":"https://pith.science/pith/2WL6FSOX3WY3VC6F5RNENKPMU3.json","view_paper":"https://pith.science/paper/2WL6FSOX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.01637&json=true","fetch_graph":"https://pith.science/api/pith-number/2WL6FSOX3WY3VC6F5RNENKPMU3/graph.json","fetch_events":"https://pith.science/api/pith-number/2WL6FSOX3WY3VC6F5RNENKPMU3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2WL6FSOX3WY3VC6F5RNENKPMU3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2WL6FSOX3WY3VC6F5RNENKPMU3/action/storage_attestation","attest_author":"https://pith.science/pith/2WL6FSOX3WY3VC6F5RNENKPMU3/action/author_attestation","sign_citation":"https://pith.science/pith/2WL6FSOX3WY3VC6F5RNENKPMU3/action/citation_signature","submit_replication":"https://pith.science/pith/2WL6FSOX3WY3VC6F5RNENKPMU3/action/replication_record"}},"created_at":"2026-05-18T01:25:10.754628+00:00","updated_at":"2026-05-18T01:25:10.754628+00:00"}