{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2004:UMTI5UCNI2DBGZYVUOBIWENNMH","short_pith_number":"pith:UMTI5UCN","canonical_record":{"source":{"id":"math/0407326","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2004-07-19T16:23:28Z","cross_cats_sorted":[],"title_canon_sha256":"79c63ed1b475577aec11235296e0fce3635251aa41c1741887ab24e5f166a3f7","abstract_canon_sha256":"1ee47123ede52358173de3d0a0f0146ab222f29c719fd3f44d5d70082fc33caf"},"schema_version":"1.0"},"canonical_sha256":"a3268ed04d4686136715a3828b11ad61ca6b9b6afc9d5cb8aba41b8fa44202fa","source":{"kind":"arxiv","id":"math/0407326","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0407326","created_at":"2026-07-04T14:38:56Z"},{"alias_kind":"arxiv_version","alias_value":"math/0407326v1","created_at":"2026-07-04T14:38:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0407326","created_at":"2026-07-04T14:38:56Z"},{"alias_kind":"pith_short_12","alias_value":"UMTI5UCNI2DB","created_at":"2026-07-04T14:38:56Z"},{"alias_kind":"pith_short_16","alias_value":"UMTI5UCNI2DBGZYV","created_at":"2026-07-04T14:38:56Z"},{"alias_kind":"pith_short_8","alias_value":"UMTI5UCN","created_at":"2026-07-04T14:38:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2004:UMTI5UCNI2DBGZYVUOBIWENNMH","target":"record","payload":{"canonical_record":{"source":{"id":"math/0407326","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2004-07-19T16:23:28Z","cross_cats_sorted":[],"title_canon_sha256":"79c63ed1b475577aec11235296e0fce3635251aa41c1741887ab24e5f166a3f7","abstract_canon_sha256":"1ee47123ede52358173de3d0a0f0146ab222f29c719fd3f44d5d70082fc33caf"},"schema_version":"1.0"},"canonical_sha256":"a3268ed04d4686136715a3828b11ad61ca6b9b6afc9d5cb8aba41b8fa44202fa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:38:56.189811Z","signature_b64":"4EsxcSsdkLFdcgaqQ7OY2ogtCC3QRa3RlA3SIPbmCsnXg52KolSVnRNJfO62AhLm3wIyWJ9Kzzu5KJhkHEDpAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3268ed04d4686136715a3828b11ad61ca6b9b6afc9d5cb8aba41b8fa44202fa","last_reissued_at":"2026-07-04T14:38:56.189465Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:38:56.189465Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0407326","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":"2026-07-04T14:38:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GKOvxgEJc8iQh6nl3kNjLx7M0xlVEBWvfA3Q+i754HlC7f1lmppLvFpSqW9vyDkOVatElUKXk38pGZkbM2vECQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T19:14:04.050611Z"},"content_sha256":"92815bf109d0a5c22117f4550356d8ec860690b97189ce29dd58e4b59ba73c6c","schema_version":"1.0","event_id":"sha256:92815bf109d0a5c22117f4550356d8ec860690b97189ce29dd58e4b59ba73c6c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2004:UMTI5UCNI2DBGZYVUOBIWENNMH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Congruences for Catalan and Motzkin numbers and related sequences","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bruce E. Sagan (Michigan State U.), Emeric Deutsch (Polytechnic U.)","submitted_at":"2004-07-19T16:23:28Z","abstract_excerpt":"We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic: group actions, induction, and Lucas' congruence for binomial coefficients come into play. A number of our results settle conjectures of Benoit Cloitre and Reinhard Zumkeller. The Thue-Morse sequence appears in several contexts."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0407326","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0407326/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T14:38:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9wGPABOuUbvcytusz4i5XGhRklAXy93RpFBHsXUGAO2Bm70PIKnenlPZymIafpx/kwpZKR0jFG9yp4BhkPtRAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T19:14:04.051579Z"},"content_sha256":"3be91137956049d697ea3b8aee61cedd20952224fd9b2522572042bd941141c6","schema_version":"1.0","event_id":"sha256:3be91137956049d697ea3b8aee61cedd20952224fd9b2522572042bd941141c6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UMTI5UCNI2DBGZYVUOBIWENNMH/bundle.json","state_url":"https://pith.science/pith/UMTI5UCNI2DBGZYVUOBIWENNMH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UMTI5UCNI2DBGZYVUOBIWENNMH/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":"2026-08-16T19:14:04Z","links":{"resolver":"https://pith.science/pith/UMTI5UCNI2DBGZYVUOBIWENNMH","bundle":"https://pith.science/pith/UMTI5UCNI2DBGZYVUOBIWENNMH/bundle.json","state":"https://pith.science/pith/UMTI5UCNI2DBGZYVUOBIWENNMH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UMTI5UCNI2DBGZYVUOBIWENNMH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:UMTI5UCNI2DBGZYVUOBIWENNMH","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":"1ee47123ede52358173de3d0a0f0146ab222f29c719fd3f44d5d70082fc33caf","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2004-07-19T16:23:28Z","title_canon_sha256":"79c63ed1b475577aec11235296e0fce3635251aa41c1741887ab24e5f166a3f7"},"schema_version":"1.0","source":{"id":"math/0407326","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0407326","created_at":"2026-07-04T14:38:56Z"},{"alias_kind":"arxiv_version","alias_value":"math/0407326v1","created_at":"2026-07-04T14:38:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0407326","created_at":"2026-07-04T14:38:56Z"},{"alias_kind":"pith_short_12","alias_value":"UMTI5UCNI2DB","created_at":"2026-07-04T14:38:56Z"},{"alias_kind":"pith_short_16","alias_value":"UMTI5UCNI2DBGZYV","created_at":"2026-07-04T14:38:56Z"},{"alias_kind":"pith_short_8","alias_value":"UMTI5UCN","created_at":"2026-07-04T14:38:56Z"}],"graph_snapshots":[{"event_id":"sha256:3be91137956049d697ea3b8aee61cedd20952224fd9b2522572042bd941141c6","target":"graph","created_at":"2026-07-04T14:38:56Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0407326/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic: group actions, induction, and Lucas' congruence for binomial coefficients come into play. A number of our results settle conjectures of Benoit Cloitre and Reinhard Zumkeller. The Thue-Morse sequence appears in several contexts.","authors_text":"Bruce E. Sagan (Michigan State U.), Emeric Deutsch (Polytechnic U.)","cross_cats":[],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2004-07-19T16:23:28Z","title":"Congruences for Catalan and Motzkin numbers and related sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0407326","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":"sha256:92815bf109d0a5c22117f4550356d8ec860690b97189ce29dd58e4b59ba73c6c","target":"record","created_at":"2026-07-04T14:38:56Z","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":"1ee47123ede52358173de3d0a0f0146ab222f29c719fd3f44d5d70082fc33caf","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2004-07-19T16:23:28Z","title_canon_sha256":"79c63ed1b475577aec11235296e0fce3635251aa41c1741887ab24e5f166a3f7"},"schema_version":"1.0","source":{"id":"math/0407326","kind":"arxiv","version":1}},"canonical_sha256":"a3268ed04d4686136715a3828b11ad61ca6b9b6afc9d5cb8aba41b8fa44202fa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a3268ed04d4686136715a3828b11ad61ca6b9b6afc9d5cb8aba41b8fa44202fa","first_computed_at":"2026-07-04T14:38:56.189465Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:38:56.189465Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4EsxcSsdkLFdcgaqQ7OY2ogtCC3QRa3RlA3SIPbmCsnXg52KolSVnRNJfO62AhLm3wIyWJ9Kzzu5KJhkHEDpAA==","signature_status":"signed_v1","signed_at":"2026-07-04T14:38:56.189811Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0407326","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:92815bf109d0a5c22117f4550356d8ec860690b97189ce29dd58e4b59ba73c6c","sha256:3be91137956049d697ea3b8aee61cedd20952224fd9b2522572042bd941141c6"],"state_sha256":"6294e52eb7946c04f105a02f4535e62cfe1c734a894c70e29443200451ededa7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2EB8T5GInrPl/76vWg70ZUcCB9jrYQZsMYFBT8zaFvo7abpzHM4IHTMGWa4nHVdKzUG/h2kxXeYsr/hd9/QCAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T19:14:04.080856Z","bundle_sha256":"18d2955a82c8a829ed69ccab1e894f46f6c565fabe059b6600a6d3798ee77e5f"}}