{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:H7U4N36DQHAFB734KZOKLZAUXZ","short_pith_number":"pith:H7U4N36D","canonical_record":{"source":{"id":"1803.06706","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-18T18:08:48Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"c45ecfe728beff866530bf3991287be2e2c785de7da93a788fec55daf3cd82f8","abstract_canon_sha256":"80e1cbeca5a4d6b98d0f01eaf650a9c1bb70cf79ffb407e1ca0e465d358922b4"},"schema_version":"1.0"},"canonical_sha256":"3fe9c6efc381c050ff7c565ca5e414be694d73f5683ee887a02b097894a65f6c","source":{"kind":"arxiv","id":"1803.06706","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.06706","created_at":"2026-05-18T00:20:45Z"},{"alias_kind":"arxiv_version","alias_value":"1803.06706v1","created_at":"2026-05-18T00:20:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.06706","created_at":"2026-05-18T00:20:45Z"},{"alias_kind":"pith_short_12","alias_value":"H7U4N36DQHAF","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"H7U4N36DQHAFB734","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"H7U4N36D","created_at":"2026-05-18T12:32:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:H7U4N36DQHAFB734KZOKLZAUXZ","target":"record","payload":{"canonical_record":{"source":{"id":"1803.06706","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-18T18:08:48Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"c45ecfe728beff866530bf3991287be2e2c785de7da93a788fec55daf3cd82f8","abstract_canon_sha256":"80e1cbeca5a4d6b98d0f01eaf650a9c1bb70cf79ffb407e1ca0e465d358922b4"},"schema_version":"1.0"},"canonical_sha256":"3fe9c6efc381c050ff7c565ca5e414be694d73f5683ee887a02b097894a65f6c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:45.214070Z","signature_b64":"xdw6L2EekVv2sg/NNi+9eIz/PhVobNNRqcr2d7cxlQ6NkW4g1tTR1K/c9h+0siDB6jrg4YqCYMLVm4DfeqmdCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3fe9c6efc381c050ff7c565ca5e414be694d73f5683ee887a02b097894a65f6c","last_reissued_at":"2026-05-18T00:20:45.213475Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:45.213475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.06706","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":"2026-05-18T00:20:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uPjqCN+++x0JKXSPmVU7xdJkaOm4s4tXZguO02bPTkRrq8qw8ItykXMcmGodp992ThSLBFfjHxuwY4wDGOgHBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T10:06:42.607751Z"},"content_sha256":"4e2e5549e42c1052810cce68f37f95bac575296abf5d492030d87a088a0d5d21","schema_version":"1.0","event_id":"sha256:4e2e5549e42c1052810cce68f37f95bac575296abf5d492030d87a088a0d5d21"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:H7U4N36DQHAFB734KZOKLZAUXZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Descent distribution on Catalan words avoiding a pattern of length at most three","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jean-Luc Baril, Sergey Kirgizov, Vincent Vajnovszki","submitted_at":"2018-03-18T18:08:48Z","abstract_excerpt":"Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern $p$ we provide a bivariate generating function where the coefficient of $x^ny^k$ in its series expansion is the number of length $n$ Catalan words with $k$ descents and avoiding $p$. As a byproduct, we enumerate the set of Catalan words avoiding $p$, and we provide the popularity of desc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.06706","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":"2026-05-18T00:20:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9ugSpjmQyOTNjzGIiqLM6kEdYP/KrsheRIEvuqxHabtNc9Vqu0OQifxlQFKc7p1YLsAgcmoFdP3axTURw4K2Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T10:06:42.608087Z"},"content_sha256":"24c909cc1a3c25fad182ad3f4e63d2fabaca67b8ab4ce8ba675205c6a886d349","schema_version":"1.0","event_id":"sha256:24c909cc1a3c25fad182ad3f4e63d2fabaca67b8ab4ce8ba675205c6a886d349"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H7U4N36DQHAFB734KZOKLZAUXZ/bundle.json","state_url":"https://pith.science/pith/H7U4N36DQHAFB734KZOKLZAUXZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H7U4N36DQHAFB734KZOKLZAUXZ/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":"2026-06-02T10:06:42Z","links":{"resolver":"https://pith.science/pith/H7U4N36DQHAFB734KZOKLZAUXZ","bundle":"https://pith.science/pith/H7U4N36DQHAFB734KZOKLZAUXZ/bundle.json","state":"https://pith.science/pith/H7U4N36DQHAFB734KZOKLZAUXZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H7U4N36DQHAFB734KZOKLZAUXZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:H7U4N36DQHAFB734KZOKLZAUXZ","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":"80e1cbeca5a4d6b98d0f01eaf650a9c1bb70cf79ffb407e1ca0e465d358922b4","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-18T18:08:48Z","title_canon_sha256":"c45ecfe728beff866530bf3991287be2e2c785de7da93a788fec55daf3cd82f8"},"schema_version":"1.0","source":{"id":"1803.06706","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.06706","created_at":"2026-05-18T00:20:45Z"},{"alias_kind":"arxiv_version","alias_value":"1803.06706v1","created_at":"2026-05-18T00:20:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.06706","created_at":"2026-05-18T00:20:45Z"},{"alias_kind":"pith_short_12","alias_value":"H7U4N36DQHAF","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"H7U4N36DQHAFB734","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"H7U4N36D","created_at":"2026-05-18T12:32:28Z"}],"graph_snapshots":[{"event_id":"sha256:24c909cc1a3c25fad182ad3f4e63d2fabaca67b8ab4ce8ba675205c6a886d349","target":"graph","created_at":"2026-05-18T00:20:45Z","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":"Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern $p$ we provide a bivariate generating function where the coefficient of $x^ny^k$ in its series expansion is the number of length $n$ Catalan words with $k$ descents and avoiding $p$. As a byproduct, we enumerate the set of Catalan words avoiding $p$, and we provide the popularity of desc","authors_text":"Jean-Luc Baril, Sergey Kirgizov, Vincent Vajnovszki","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-18T18:08:48Z","title":"Descent distribution on Catalan words avoiding a pattern of length at most three"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.06706","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":"sha256:4e2e5549e42c1052810cce68f37f95bac575296abf5d492030d87a088a0d5d21","target":"record","created_at":"2026-05-18T00:20:45Z","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":"80e1cbeca5a4d6b98d0f01eaf650a9c1bb70cf79ffb407e1ca0e465d358922b4","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-18T18:08:48Z","title_canon_sha256":"c45ecfe728beff866530bf3991287be2e2c785de7da93a788fec55daf3cd82f8"},"schema_version":"1.0","source":{"id":"1803.06706","kind":"arxiv","version":1}},"canonical_sha256":"3fe9c6efc381c050ff7c565ca5e414be694d73f5683ee887a02b097894a65f6c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3fe9c6efc381c050ff7c565ca5e414be694d73f5683ee887a02b097894a65f6c","first_computed_at":"2026-05-18T00:20:45.213475Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:45.213475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xdw6L2EekVv2sg/NNi+9eIz/PhVobNNRqcr2d7cxlQ6NkW4g1tTR1K/c9h+0siDB6jrg4YqCYMLVm4DfeqmdCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:45.214070Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.06706","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4e2e5549e42c1052810cce68f37f95bac575296abf5d492030d87a088a0d5d21","sha256:24c909cc1a3c25fad182ad3f4e63d2fabaca67b8ab4ce8ba675205c6a886d349"],"state_sha256":"c07a989390b581a4a9929f063fd8f527772bdb060f6d4073a0886d079a2e34bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pDo6AeB8JZaDpR4iInJGF49mH5VzkteEphz/AKkGDkLxXzVwVoX6vOuIH0cNyXEdsbhxmdzkJfMVpK5BnmksDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T10:06:42.609904Z","bundle_sha256":"78e25204631efcfd4a91118b659bf7cad9c59a7b676fae26ec39d675884e5ec2"}}