{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:BWQI5OY7TFULKZ4IHAZRLBE42S","short_pith_number":"pith:BWQI5OY7","canonical_record":{"source":{"id":"1904.09748","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-22T07:04:42Z","cross_cats_sorted":[],"title_canon_sha256":"a6a5bb1085b7e43d854b41f1ecd5abd96370960646f8062159239a2a028bbe48","abstract_canon_sha256":"e0c30c0e4b29c919c3bf8da900d588e369b4360db758ba70a72fc4ef272839de"},"schema_version":"1.0"},"canonical_sha256":"0da08ebb1f9968b56788383315849cd4aa4b0d1b073b3081f9774601ba0f83e4","source":{"kind":"arxiv","id":"1904.09748","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.09748","created_at":"2026-07-05T03:30:05Z"},{"alias_kind":"arxiv_version","alias_value":"1904.09748v4","created_at":"2026-07-05T03:30:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.09748","created_at":"2026-07-05T03:30:05Z"},{"alias_kind":"pith_short_12","alias_value":"BWQI5OY7TFUL","created_at":"2026-07-05T03:30:05Z"},{"alias_kind":"pith_short_16","alias_value":"BWQI5OY7TFULKZ4I","created_at":"2026-07-05T03:30:05Z"},{"alias_kind":"pith_short_8","alias_value":"BWQI5OY7","created_at":"2026-07-05T03:30:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:BWQI5OY7TFULKZ4IHAZRLBE42S","target":"record","payload":{"canonical_record":{"source":{"id":"1904.09748","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-22T07:04:42Z","cross_cats_sorted":[],"title_canon_sha256":"a6a5bb1085b7e43d854b41f1ecd5abd96370960646f8062159239a2a028bbe48","abstract_canon_sha256":"e0c30c0e4b29c919c3bf8da900d588e369b4360db758ba70a72fc4ef272839de"},"schema_version":"1.0"},"canonical_sha256":"0da08ebb1f9968b56788383315849cd4aa4b0d1b073b3081f9774601ba0f83e4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:30:05.499805Z","signature_b64":"YaHhAX/6DwuZH5BqP6cP2fvyHd7FdLdPDHhMgSaoKW/nOUaKeDObBLV11+/mjlgr8juJE3W/88DqfaDdHFsTAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0da08ebb1f9968b56788383315849cd4aa4b0d1b073b3081f9774601ba0f83e4","last_reissued_at":"2026-07-05T03:30:05.499329Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:30:05.499329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.09748","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:30:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lTlLTmGRmXHjZGnpXWoxTDazBvFRmgkDNX03Z7ngo6HSmEg8mAphHJHpXtxCn2wy1iWmorlm1uzblPsVnGZ3Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T12:33:22.379753Z"},"content_sha256":"1edcb2e3620d37938e52a782a285f0c4a39126ab5809ebb8140e496b0e5028e3","schema_version":"1.0","event_id":"sha256:1edcb2e3620d37938e52a782a285f0c4a39126ab5809ebb8140e496b0e5028e3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:BWQI5OY7TFULKZ4IHAZRLBE42S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Norihiro Nakashima, Shuhei Tsujie","submitted_at":"2019-04-22T07:04:42Z","abstract_excerpt":"The number of flats of a hyperplane arrangement is considered as a generalization of the Bell number and the Stirling number of the second kind. Robert Gill gave the exponential generating function of the number of flats of the extended Catalan arrangements, using species. In this article, we introduce the species of flats of the extended Catalan and Shi arrangements and they are given by iterated substitution of species of sets and lists. Moreover, we enumerate the flats of these arrangements in terms of infinite matrices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.09748","kind":"arxiv","version":4},"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/1904.09748/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":"2026-07-05T03:30:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p+Oc9p99FMTL3w+rCXkyGEKNdb5fFjO5rjgpNRD4vGKkVE5O46tLj0/n4urVXlzXqX86C7C/6RqJ9Ld46OSeDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T12:33:22.380390Z"},"content_sha256":"6aa0e912f8673a7f0b8b931872306004176da94d359e3feb041b1bda8616a23b","schema_version":"1.0","event_id":"sha256:6aa0e912f8673a7f0b8b931872306004176da94d359e3feb041b1bda8616a23b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BWQI5OY7TFULKZ4IHAZRLBE42S/bundle.json","state_url":"https://pith.science/pith/BWQI5OY7TFULKZ4IHAZRLBE42S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BWQI5OY7TFULKZ4IHAZRLBE42S/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":"2026-08-16T12:33:22Z","links":{"resolver":"https://pith.science/pith/BWQI5OY7TFULKZ4IHAZRLBE42S","bundle":"https://pith.science/pith/BWQI5OY7TFULKZ4IHAZRLBE42S/bundle.json","state":"https://pith.science/pith/BWQI5OY7TFULKZ4IHAZRLBE42S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BWQI5OY7TFULKZ4IHAZRLBE42S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:BWQI5OY7TFULKZ4IHAZRLBE42S","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":"e0c30c0e4b29c919c3bf8da900d588e369b4360db758ba70a72fc4ef272839de","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-22T07:04:42Z","title_canon_sha256":"a6a5bb1085b7e43d854b41f1ecd5abd96370960646f8062159239a2a028bbe48"},"schema_version":"1.0","source":{"id":"1904.09748","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.09748","created_at":"2026-07-05T03:30:05Z"},{"alias_kind":"arxiv_version","alias_value":"1904.09748v4","created_at":"2026-07-05T03:30:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.09748","created_at":"2026-07-05T03:30:05Z"},{"alias_kind":"pith_short_12","alias_value":"BWQI5OY7TFUL","created_at":"2026-07-05T03:30:05Z"},{"alias_kind":"pith_short_16","alias_value":"BWQI5OY7TFULKZ4I","created_at":"2026-07-05T03:30:05Z"},{"alias_kind":"pith_short_8","alias_value":"BWQI5OY7","created_at":"2026-07-05T03:30:05Z"}],"graph_snapshots":[{"event_id":"sha256:6aa0e912f8673a7f0b8b931872306004176da94d359e3feb041b1bda8616a23b","target":"graph","created_at":"2026-07-05T03:30:05Z","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/1904.09748/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The number of flats of a hyperplane arrangement is considered as a generalization of the Bell number and the Stirling number of the second kind. Robert Gill gave the exponential generating function of the number of flats of the extended Catalan arrangements, using species. In this article, we introduce the species of flats of the extended Catalan and Shi arrangements and they are given by iterated substitution of species of sets and lists. Moreover, we enumerate the flats of these arrangements in terms of infinite matrices.","authors_text":"Norihiro Nakashima, Shuhei Tsujie","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-22T07:04:42Z","title":"Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.09748","kind":"arxiv","version":4},"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":"sha256:1edcb2e3620d37938e52a782a285f0c4a39126ab5809ebb8140e496b0e5028e3","target":"record","created_at":"2026-07-05T03:30:05Z","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":"e0c30c0e4b29c919c3bf8da900d588e369b4360db758ba70a72fc4ef272839de","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-22T07:04:42Z","title_canon_sha256":"a6a5bb1085b7e43d854b41f1ecd5abd96370960646f8062159239a2a028bbe48"},"schema_version":"1.0","source":{"id":"1904.09748","kind":"arxiv","version":4}},"canonical_sha256":"0da08ebb1f9968b56788383315849cd4aa4b0d1b073b3081f9774601ba0f83e4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0da08ebb1f9968b56788383315849cd4aa4b0d1b073b3081f9774601ba0f83e4","first_computed_at":"2026-07-05T03:30:05.499329Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:30:05.499329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YaHhAX/6DwuZH5BqP6cP2fvyHd7FdLdPDHhMgSaoKW/nOUaKeDObBLV11+/mjlgr8juJE3W/88DqfaDdHFsTAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:30:05.499805Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.09748","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1edcb2e3620d37938e52a782a285f0c4a39126ab5809ebb8140e496b0e5028e3","sha256:6aa0e912f8673a7f0b8b931872306004176da94d359e3feb041b1bda8616a23b"],"state_sha256":"ccd0717dba16bdee190e5b0d4fd9234a013450d784351be150e81fcf91dda07f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s3QBcyMYU7Bw7rYzIz0ozgcbsy+ErimiVvx2gz4BVYOOq4D65COzD7EI4oeJ4vfE225AdjWCcGAUDxwcNa61Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T12:33:22.385857Z","bundle_sha256":"5e5ab551c8029ab3e4df9eca32ed2bd4e80a7442ff673634387bbf4544987d85"}}