{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:VVSGPBKOZF4UB5E43CRZIKSHW3","short_pith_number":"pith:VVSGPBKO","canonical_record":{"source":{"id":"1711.08584","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-23T06:16:13Z","cross_cats_sorted":[],"title_canon_sha256":"23bc0f8205c165a474503875b4fc54f623ef7b65578aa2eca4aec7bd12d08a51","abstract_canon_sha256":"560f730ff0252e42e14474fb8df484c36e190e17cab437863005a5fcefdb6f2a"},"schema_version":"1.0"},"canonical_sha256":"ad6467854ec97940f49cd8a3942a47b6cf6d06604982490dacfdc37b269f56e0","source":{"kind":"arxiv","id":"1711.08584","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.08584","created_at":"2026-05-18T00:24:59Z"},{"alias_kind":"arxiv_version","alias_value":"1711.08584v2","created_at":"2026-05-18T00:24:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.08584","created_at":"2026-05-18T00:24:59Z"},{"alias_kind":"pith_short_12","alias_value":"VVSGPBKOZF4U","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"VVSGPBKOZF4UB5E4","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"VVSGPBKO","created_at":"2026-05-18T12:31:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:VVSGPBKOZF4UB5E43CRZIKSHW3","target":"record","payload":{"canonical_record":{"source":{"id":"1711.08584","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-23T06:16:13Z","cross_cats_sorted":[],"title_canon_sha256":"23bc0f8205c165a474503875b4fc54f623ef7b65578aa2eca4aec7bd12d08a51","abstract_canon_sha256":"560f730ff0252e42e14474fb8df484c36e190e17cab437863005a5fcefdb6f2a"},"schema_version":"1.0"},"canonical_sha256":"ad6467854ec97940f49cd8a3942a47b6cf6d06604982490dacfdc37b269f56e0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:59.375634Z","signature_b64":"tH2a7uhQ2S4Z/+Ks0BgCpAy1l7j0KTBAuNZy1PeFEBcoIaDKG/DiDcsx+Q65PmKsZEakOQfZsjdiBwly6epnAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad6467854ec97940f49cd8a3942a47b6cf6d06604982490dacfdc37b269f56e0","last_reissued_at":"2026-05-18T00:24:59.375256Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:59.375256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1711.08584","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:24:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ModWrUSV81WlhOk6FvTNg4UtgYkJ+5W9viKauj7ns8xMTOlGj3xPfH/43CAF6NuffdegyQSGyQC8zH0TyeWnBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:32:02.026094Z"},"content_sha256":"e6b8fe57278e4a5b4cc08437c28c0c8d9fd272104cbe54b9b83e5caffb40e10c","schema_version":"1.0","event_id":"sha256:e6b8fe57278e4a5b4cc08437c28c0c8d9fd272104cbe54b9b83e5caffb40e10c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:VVSGPBKOZF4UB5E43CRZIKSHW3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Refinements of two identities on $(n,m)$-Dyck paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kuo Yu, Rosena R. X. Du","submitted_at":"2017-11-23T06:16:13Z","abstract_excerpt":"For integers $n, m$ with $n \\geq 1$ and $0 \\leq m \\leq n$, an $(n,m)$-Dyck path is a lattice path in the integer lattice $\\mathbb{Z} \\times \\mathbb{Z}$ using up steps $(0,1)$ and down steps $(1,0)$ that goes from the origin $(0,0)$ to the point $(n,n)$ and contains exactly $m$ up steps below the line $y=x$. The classical Chung-Feller theorem says that the total number of $(n,m)$-Dyck path is independent of $m$ and is equal to the $n$-th Catalan number $C_n=\\frac{1}{n+1}{2n \\choose n}$. For any integer $k$ with $1 \\leq k \\leq n$, let $p_{n,m,k}$ be the total number of $(n,m)$-Dyck paths with $k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.08584","kind":"arxiv","version":2},"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":"2026-05-18T00:24:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oNFqzzgqE90gTUmm3VTBikkKjt/sp6PlEV6W9Az0E5bViQHUYxyKfaU58+6bbsmMTFHx50843AvYRT5v3GhODQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:32:02.026463Z"},"content_sha256":"f96bf6028880f66429d16af25cc0bdb874f552481bb5b32b5f0a5344066cdb12","schema_version":"1.0","event_id":"sha256:f96bf6028880f66429d16af25cc0bdb874f552481bb5b32b5f0a5344066cdb12"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VVSGPBKOZF4UB5E43CRZIKSHW3/bundle.json","state_url":"https://pith.science/pith/VVSGPBKOZF4UB5E43CRZIKSHW3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VVSGPBKOZF4UB5E43CRZIKSHW3/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":"2026-06-03T04:32:02Z","links":{"resolver":"https://pith.science/pith/VVSGPBKOZF4UB5E43CRZIKSHW3","bundle":"https://pith.science/pith/VVSGPBKOZF4UB5E43CRZIKSHW3/bundle.json","state":"https://pith.science/pith/VVSGPBKOZF4UB5E43CRZIKSHW3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VVSGPBKOZF4UB5E43CRZIKSHW3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:VVSGPBKOZF4UB5E43CRZIKSHW3","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":"560f730ff0252e42e14474fb8df484c36e190e17cab437863005a5fcefdb6f2a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-23T06:16:13Z","title_canon_sha256":"23bc0f8205c165a474503875b4fc54f623ef7b65578aa2eca4aec7bd12d08a51"},"schema_version":"1.0","source":{"id":"1711.08584","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.08584","created_at":"2026-05-18T00:24:59Z"},{"alias_kind":"arxiv_version","alias_value":"1711.08584v2","created_at":"2026-05-18T00:24:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.08584","created_at":"2026-05-18T00:24:59Z"},{"alias_kind":"pith_short_12","alias_value":"VVSGPBKOZF4U","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"VVSGPBKOZF4UB5E4","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"VVSGPBKO","created_at":"2026-05-18T12:31:49Z"}],"graph_snapshots":[{"event_id":"sha256:f96bf6028880f66429d16af25cc0bdb874f552481bb5b32b5f0a5344066cdb12","target":"graph","created_at":"2026-05-18T00:24:59Z","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":"For integers $n, m$ with $n \\geq 1$ and $0 \\leq m \\leq n$, an $(n,m)$-Dyck path is a lattice path in the integer lattice $\\mathbb{Z} \\times \\mathbb{Z}$ using up steps $(0,1)$ and down steps $(1,0)$ that goes from the origin $(0,0)$ to the point $(n,n)$ and contains exactly $m$ up steps below the line $y=x$. The classical Chung-Feller theorem says that the total number of $(n,m)$-Dyck path is independent of $m$ and is equal to the $n$-th Catalan number $C_n=\\frac{1}{n+1}{2n \\choose n}$. For any integer $k$ with $1 \\leq k \\leq n$, let $p_{n,m,k}$ be the total number of $(n,m)$-Dyck paths with $k","authors_text":"Kuo Yu, Rosena R. X. Du","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-23T06:16:13Z","title":"Refinements of two identities on $(n,m)$-Dyck paths"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.08584","kind":"arxiv","version":2},"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:e6b8fe57278e4a5b4cc08437c28c0c8d9fd272104cbe54b9b83e5caffb40e10c","target":"record","created_at":"2026-05-18T00:24:59Z","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":"560f730ff0252e42e14474fb8df484c36e190e17cab437863005a5fcefdb6f2a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-23T06:16:13Z","title_canon_sha256":"23bc0f8205c165a474503875b4fc54f623ef7b65578aa2eca4aec7bd12d08a51"},"schema_version":"1.0","source":{"id":"1711.08584","kind":"arxiv","version":2}},"canonical_sha256":"ad6467854ec97940f49cd8a3942a47b6cf6d06604982490dacfdc37b269f56e0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ad6467854ec97940f49cd8a3942a47b6cf6d06604982490dacfdc37b269f56e0","first_computed_at":"2026-05-18T00:24:59.375256Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:24:59.375256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tH2a7uhQ2S4Z/+Ks0BgCpAy1l7j0KTBAuNZy1PeFEBcoIaDKG/DiDcsx+Q65PmKsZEakOQfZsjdiBwly6epnAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:24:59.375634Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.08584","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e6b8fe57278e4a5b4cc08437c28c0c8d9fd272104cbe54b9b83e5caffb40e10c","sha256:f96bf6028880f66429d16af25cc0bdb874f552481bb5b32b5f0a5344066cdb12"],"state_sha256":"f547ff48b51c98a5d3a09a53c97ec9b3e7eeae8d745a1d03f86aabf61d64bd3f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CxcMAA8uysp5b7Bh74hnIhSFWHVNhiU30A57zYqlZ8oESPVkMcir+pz1ZNXlvmn7N7R1IzSBa7m23ZR0qCi9Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T04:32:02.028409Z","bundle_sha256":"d40d33076dc81f10e5cbda05d1e3c6e47b8195553c17d9cfa338e64127b5a251"}}