{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2001:D2LYJIR5V5EAI2SNRNX4TCNZOZ","short_pith_number":"pith:D2LYJIR5","canonical_record":{"source":{"id":"math/0109147","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2001-09-20T18:10:37Z","cross_cats_sorted":["math.AC","math.RA"],"title_canon_sha256":"a4c406d11578ea68d002c9764c113ccf30ff5720bf556218dc956fcdd69ff824","abstract_canon_sha256":"5e2faddcc2e3804a496481717913a9a64aa519dda0e50f74661062875c5727c2"},"schema_version":"1.0"},"canonical_sha256":"1e9784a23daf48046a4d8b6fc989b976730af5953373378f59d6dbf8bbbb8e67","source":{"kind":"arxiv","id":"math/0109147","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0109147","created_at":"2026-05-18T01:00:03Z"},{"alias_kind":"arxiv_version","alias_value":"math/0109147v1","created_at":"2026-05-18T01:00:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0109147","created_at":"2026-05-18T01:00:03Z"},{"alias_kind":"pith_short_12","alias_value":"D2LYJIR5V5EA","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"D2LYJIR5V5EAI2SN","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"D2LYJIR5","created_at":"2026-05-18T12:25:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2001:D2LYJIR5V5EAI2SNRNX4TCNZOZ","target":"record","payload":{"canonical_record":{"source":{"id":"math/0109147","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2001-09-20T18:10:37Z","cross_cats_sorted":["math.AC","math.RA"],"title_canon_sha256":"a4c406d11578ea68d002c9764c113ccf30ff5720bf556218dc956fcdd69ff824","abstract_canon_sha256":"5e2faddcc2e3804a496481717913a9a64aa519dda0e50f74661062875c5727c2"},"schema_version":"1.0"},"canonical_sha256":"1e9784a23daf48046a4d8b6fc989b976730af5953373378f59d6dbf8bbbb8e67","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:03.485075Z","signature_b64":"WsAqz3sYB/EZ1znSv3ZvjOEduZeVF3wfBeTvrcMjFBmD/0wKp6aJMMGbyAcVk8e9FCFDGZuJrVYydNS+vWOlCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e9784a23daf48046a4d8b6fc989b976730af5953373378f59d6dbf8bbbb8e67","last_reissued_at":"2026-05-18T01:00:03.484508Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:03.484508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0109147","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:00:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vODpuULgzaXa9YyIhlm091cgqIseKnqMyBMRj4HmVzlb5SpVd63BmWMzCGSyZ1PnzWxKqjAJZNHRoZSRtm2JCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T21:24:13.990427Z"},"content_sha256":"3ead13b2f974563f769671c17d05a4ba1d9446335c1c1bf52e47dcef7b3fbac3","schema_version":"1.0","event_id":"sha256:3ead13b2f974563f769671c17d05a4ba1d9446335c1c1bf52e47dcef7b3fbac3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2001:D2LYJIR5V5EAI2SNRNX4TCNZOZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Catalan paths, Quasi-symmetric functions and Super-Harmonic Spaces","license":"","headline":"","cross_cats":["math.AC","math.RA"],"primary_cat":"math.CO","authors_text":"Jean-Christophe Aval, Nantel Bergeron","submitted_at":"2001-09-20T18:10:37Z","abstract_excerpt":"We investigate the quotient ring $R$ of the ring of formal power series $\\Q[[x_1,x_2,...]]$ over the closure of the ideal generated by non-constant quasi-\\break symmetric functions. We show that a Hilbert basis of the quotient is naturally indexed by Catalan paths (infinite Dyck paths). We also give a filtration of ideals related to Catalan paths from $(0,0)$ and above the line $y=x-k$. We investigate as well the quotient ring $R_n$ of polynomial ring in $n$ variables over the ideal generated by non-constant quasi-symmetric polynomials. We show that the dimension of $R_n$ is bounded above by t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0109147","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:00:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Tq6HsQ4sNVZOJGi2+fpFnM/MXXStJfaIKManWL8EUUiJstYqBXI1BQs693fQph/UiLqbfnKMbfISETNV0TfBDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T21:24:13.990785Z"},"content_sha256":"efdc6a3b255ecc2f90c9ba5a61b83e8f9b7a6bac7a5e05426fd07d377378bc3d","schema_version":"1.0","event_id":"sha256:efdc6a3b255ecc2f90c9ba5a61b83e8f9b7a6bac7a5e05426fd07d377378bc3d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/D2LYJIR5V5EAI2SNRNX4TCNZOZ/bundle.json","state_url":"https://pith.science/pith/D2LYJIR5V5EAI2SNRNX4TCNZOZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/D2LYJIR5V5EAI2SNRNX4TCNZOZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-24T21:24:13Z","links":{"resolver":"https://pith.science/pith/D2LYJIR5V5EAI2SNRNX4TCNZOZ","bundle":"https://pith.science/pith/D2LYJIR5V5EAI2SNRNX4TCNZOZ/bundle.json","state":"https://pith.science/pith/D2LYJIR5V5EAI2SNRNX4TCNZOZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/D2LYJIR5V5EAI2SNRNX4TCNZOZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2001:D2LYJIR5V5EAI2SNRNX4TCNZOZ","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":"5e2faddcc2e3804a496481717913a9a64aa519dda0e50f74661062875c5727c2","cross_cats_sorted":["math.AC","math.RA"],"license":"","primary_cat":"math.CO","submitted_at":"2001-09-20T18:10:37Z","title_canon_sha256":"a4c406d11578ea68d002c9764c113ccf30ff5720bf556218dc956fcdd69ff824"},"schema_version":"1.0","source":{"id":"math/0109147","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0109147","created_at":"2026-05-18T01:00:03Z"},{"alias_kind":"arxiv_version","alias_value":"math/0109147v1","created_at":"2026-05-18T01:00:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0109147","created_at":"2026-05-18T01:00:03Z"},{"alias_kind":"pith_short_12","alias_value":"D2LYJIR5V5EA","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"D2LYJIR5V5EAI2SN","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"D2LYJIR5","created_at":"2026-05-18T12:25:50Z"}],"graph_snapshots":[{"event_id":"sha256:efdc6a3b255ecc2f90c9ba5a61b83e8f9b7a6bac7a5e05426fd07d377378bc3d","target":"graph","created_at":"2026-05-18T01:00:03Z","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"},"paper":{"abstract_excerpt":"We investigate the quotient ring $R$ of the ring of formal power series $\\Q[[x_1,x_2,...]]$ over the closure of the ideal generated by non-constant quasi-\\break symmetric functions. We show that a Hilbert basis of the quotient is naturally indexed by Catalan paths (infinite Dyck paths). We also give a filtration of ideals related to Catalan paths from $(0,0)$ and above the line $y=x-k$. We investigate as well the quotient ring $R_n$ of polynomial ring in $n$ variables over the ideal generated by non-constant quasi-symmetric polynomials. We show that the dimension of $R_n$ is bounded above by t","authors_text":"Jean-Christophe Aval, Nantel Bergeron","cross_cats":["math.AC","math.RA"],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2001-09-20T18:10:37Z","title":"Catalan paths, Quasi-symmetric functions and Super-Harmonic Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0109147","kind":"arxiv","version":1},"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":"sha256:3ead13b2f974563f769671c17d05a4ba1d9446335c1c1bf52e47dcef7b3fbac3","target":"record","created_at":"2026-05-18T01:00:03Z","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":"5e2faddcc2e3804a496481717913a9a64aa519dda0e50f74661062875c5727c2","cross_cats_sorted":["math.AC","math.RA"],"license":"","primary_cat":"math.CO","submitted_at":"2001-09-20T18:10:37Z","title_canon_sha256":"a4c406d11578ea68d002c9764c113ccf30ff5720bf556218dc956fcdd69ff824"},"schema_version":"1.0","source":{"id":"math/0109147","kind":"arxiv","version":1}},"canonical_sha256":"1e9784a23daf48046a4d8b6fc989b976730af5953373378f59d6dbf8bbbb8e67","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1e9784a23daf48046a4d8b6fc989b976730af5953373378f59d6dbf8bbbb8e67","first_computed_at":"2026-05-18T01:00:03.484508Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:00:03.484508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WsAqz3sYB/EZ1znSv3ZvjOEduZeVF3wfBeTvrcMjFBmD/0wKp6aJMMGbyAcVk8e9FCFDGZuJrVYydNS+vWOlCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:00:03.485075Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0109147","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3ead13b2f974563f769671c17d05a4ba1d9446335c1c1bf52e47dcef7b3fbac3","sha256:efdc6a3b255ecc2f90c9ba5a61b83e8f9b7a6bac7a5e05426fd07d377378bc3d"],"state_sha256":"fcca8636ba93af94bc1623e6e2121353c441ac1cdfd6c684b413764ab7c12f59"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TIzZ9qq9RAnFq8ezldEockPiGSlRom2UP25Nu5nQ+GbwayDWbtgj9qj5THKZoLLrtgdy4eYQDpSFPHQWWDTgDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T21:24:13.992764Z","bundle_sha256":"d11bef9496712f1e89f7bb36d57fb1735ecbe8ab5167bade97fc8faf53fc2d00"}}