{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:ECNAS6XW66CSDYT76EJRBMT3RN","short_pith_number":"pith:ECNAS6XW","canonical_record":{"source":{"id":"1705.08343","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-23T15:03:04Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"3571d4856feb041e327ba532254793a143a70347b6b2b57658a76f04e5431056","abstract_canon_sha256":"02817e72256992a524bcc7dc8b4a9fe2991a0f0517f9ea039cd345a67c1018d0"},"schema_version":"1.0"},"canonical_sha256":"209a097af6f78521e27ff11310b27b8b701a72096d0c15d7dd193894e9aab50d","source":{"kind":"arxiv","id":"1705.08343","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.08343","created_at":"2026-05-18T00:43:48Z"},{"alias_kind":"arxiv_version","alias_value":"1705.08343v1","created_at":"2026-05-18T00:43:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.08343","created_at":"2026-05-18T00:43:48Z"},{"alias_kind":"pith_short_12","alias_value":"ECNAS6XW66CS","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_16","alias_value":"ECNAS6XW66CSDYT7","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_8","alias_value":"ECNAS6XW","created_at":"2026-05-18T12:31:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:ECNAS6XW66CSDYT76EJRBMT3RN","target":"record","payload":{"canonical_record":{"source":{"id":"1705.08343","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-23T15:03:04Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"3571d4856feb041e327ba532254793a143a70347b6b2b57658a76f04e5431056","abstract_canon_sha256":"02817e72256992a524bcc7dc8b4a9fe2991a0f0517f9ea039cd345a67c1018d0"},"schema_version":"1.0"},"canonical_sha256":"209a097af6f78521e27ff11310b27b8b701a72096d0c15d7dd193894e9aab50d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:48.518687Z","signature_b64":"eGoz8HbQOuVxGYD36LhmvCriDzsmugBOIsBSZJKzdA529YDo1Xl7VUABqE1zdqCzsvhB4X26LHThWaNAL6FTBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"209a097af6f78521e27ff11310b27b8b701a72096d0c15d7dd193894e9aab50d","last_reissued_at":"2026-05-18T00:43:48.517990Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:48.517990Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1705.08343","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":"2026-05-18T00:43:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f3LJ3mkdpT3WG9tyCU+PNLSrUTJEk6vPYgu8neyc6YSfQR8l+l/B2Bd02RTfxGuaHYaOfQL9nW/Akoyt/jZLDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T18:19:39.606053Z"},"content_sha256":"62fbe3c48368167450c398b6513b4511e37f198994afda0a2e51ac962b293e1b","schema_version":"1.0","event_id":"sha256:62fbe3c48368167450c398b6513b4511e37f198994afda0a2e51ac962b293e1b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:ECNAS6XW66CSDYT76EJRBMT3RN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting the number of non-zero coefficients in rows of generalized Pascal triangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Julien Leroy, Manon Stipulanti, Michel Rigo","submitted_at":"2017-05-23T15:03:04Z","abstract_excerpt":"This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence $(S(n))_{n\\ge 0}$ counting the number of positive entries on each row. By introducing a convenient tree structure, we provide a recurrence relation for $(S(n))_{n\\ge 0}$. This leads to a connection with the $2$-regular Stern-Brocot sequence and the sequence of denominators occurring in the Farey tree. Then we extend our construction to the Zeckendorf numeration system based on"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.08343","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":"2026-05-18T00:43:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dnwfxzvIzHmjNoQ5MmPhOGXigKSzuID7ts/KBqZOd5xcJqZL9estiw2zKAKvsO6vSj2MdiD0FNdPHCS+46uYBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T18:19:39.606764Z"},"content_sha256":"450a083f21a5b10d543451ecb06e5bc3b2e8c251c66f1103c40b1c9489531482","schema_version":"1.0","event_id":"sha256:450a083f21a5b10d543451ecb06e5bc3b2e8c251c66f1103c40b1c9489531482"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ECNAS6XW66CSDYT76EJRBMT3RN/bundle.json","state_url":"https://pith.science/pith/ECNAS6XW66CSDYT76EJRBMT3RN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ECNAS6XW66CSDYT76EJRBMT3RN/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":"2026-05-29T18:19:39Z","links":{"resolver":"https://pith.science/pith/ECNAS6XW66CSDYT76EJRBMT3RN","bundle":"https://pith.science/pith/ECNAS6XW66CSDYT76EJRBMT3RN/bundle.json","state":"https://pith.science/pith/ECNAS6XW66CSDYT76EJRBMT3RN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ECNAS6XW66CSDYT76EJRBMT3RN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:ECNAS6XW66CSDYT76EJRBMT3RN","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":"02817e72256992a524bcc7dc8b4a9fe2991a0f0517f9ea039cd345a67c1018d0","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-23T15:03:04Z","title_canon_sha256":"3571d4856feb041e327ba532254793a143a70347b6b2b57658a76f04e5431056"},"schema_version":"1.0","source":{"id":"1705.08343","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.08343","created_at":"2026-05-18T00:43:48Z"},{"alias_kind":"arxiv_version","alias_value":"1705.08343v1","created_at":"2026-05-18T00:43:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.08343","created_at":"2026-05-18T00:43:48Z"},{"alias_kind":"pith_short_12","alias_value":"ECNAS6XW66CS","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_16","alias_value":"ECNAS6XW66CSDYT7","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_8","alias_value":"ECNAS6XW","created_at":"2026-05-18T12:31:12Z"}],"graph_snapshots":[{"event_id":"sha256:450a083f21a5b10d543451ecb06e5bc3b2e8c251c66f1103c40b1c9489531482","target":"graph","created_at":"2026-05-18T00:43:48Z","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":"This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence $(S(n))_{n\\ge 0}$ counting the number of positive entries on each row. By introducing a convenient tree structure, we provide a recurrence relation for $(S(n))_{n\\ge 0}$. This leads to a connection with the $2$-regular Stern-Brocot sequence and the sequence of denominators occurring in the Farey tree. Then we extend our construction to the Zeckendorf numeration system based on","authors_text":"Julien Leroy, Manon Stipulanti, Michel Rigo","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-23T15:03:04Z","title":"Counting the number of non-zero coefficients in rows of generalized Pascal triangles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.08343","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":"sha256:62fbe3c48368167450c398b6513b4511e37f198994afda0a2e51ac962b293e1b","target":"record","created_at":"2026-05-18T00:43:48Z","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":"02817e72256992a524bcc7dc8b4a9fe2991a0f0517f9ea039cd345a67c1018d0","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-23T15:03:04Z","title_canon_sha256":"3571d4856feb041e327ba532254793a143a70347b6b2b57658a76f04e5431056"},"schema_version":"1.0","source":{"id":"1705.08343","kind":"arxiv","version":1}},"canonical_sha256":"209a097af6f78521e27ff11310b27b8b701a72096d0c15d7dd193894e9aab50d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"209a097af6f78521e27ff11310b27b8b701a72096d0c15d7dd193894e9aab50d","first_computed_at":"2026-05-18T00:43:48.517990Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:43:48.517990Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eGoz8HbQOuVxGYD36LhmvCriDzsmugBOIsBSZJKzdA529YDo1Xl7VUABqE1zdqCzsvhB4X26LHThWaNAL6FTBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:43:48.518687Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.08343","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:62fbe3c48368167450c398b6513b4511e37f198994afda0a2e51ac962b293e1b","sha256:450a083f21a5b10d543451ecb06e5bc3b2e8c251c66f1103c40b1c9489531482"],"state_sha256":"7401353958d4a897f248cac38ddff87179592ce05c947c1b05080d54b8a3e0ff"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T7qtvHl4X1WPI3dTEz0o4d3aUILRDGMmjpqb165Gop3Lif8CwAHM2w021VilMBgMyOhTIOp+G+hhBdsbtP5SDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T18:19:39.610601Z","bundle_sha256":"24fabcbb5b63fcb6a76bb449a4e0fc56a95703545d71130df4868bdb89019ea0"}}