{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:WEDBFB5KDIKFSWJY3TOMONO7HR","short_pith_number":"pith:WEDBFB5K","canonical_record":{"source":{"id":"1806.06417","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-17T17:34:52Z","cross_cats_sorted":[],"title_canon_sha256":"c7dd4b315f62cc23fe5a3f8101e4079010dbe2d6e2e73452156abd5ef523c208","abstract_canon_sha256":"28d27df858109503890cbcf9e60819294e9336cb5063f79f7f71f8c408ec8c6c"},"schema_version":"1.0"},"canonical_sha256":"b1061287aa1a14595938dcdcc735df3c68479aa91ba559498f55d24405567bdb","source":{"kind":"arxiv","id":"1806.06417","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.06417","created_at":"2026-07-05T04:06:39Z"},{"alias_kind":"arxiv_version","alias_value":"1806.06417v1","created_at":"2026-07-05T04:06:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.06417","created_at":"2026-07-05T04:06:39Z"},{"alias_kind":"pith_short_12","alias_value":"WEDBFB5KDIKF","created_at":"2026-07-05T04:06:39Z"},{"alias_kind":"pith_short_16","alias_value":"WEDBFB5KDIKFSWJY","created_at":"2026-07-05T04:06:39Z"},{"alias_kind":"pith_short_8","alias_value":"WEDBFB5K","created_at":"2026-07-05T04:06:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:WEDBFB5KDIKFSWJY3TOMONO7HR","target":"record","payload":{"canonical_record":{"source":{"id":"1806.06417","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-17T17:34:52Z","cross_cats_sorted":[],"title_canon_sha256":"c7dd4b315f62cc23fe5a3f8101e4079010dbe2d6e2e73452156abd5ef523c208","abstract_canon_sha256":"28d27df858109503890cbcf9e60819294e9336cb5063f79f7f71f8c408ec8c6c"},"schema_version":"1.0"},"canonical_sha256":"b1061287aa1a14595938dcdcc735df3c68479aa91ba559498f55d24405567bdb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:06:39.043463Z","signature_b64":"R6p9YjM6DHrGV5tUVB9zo69ejMcaEtYG2lt57N9/n7PnOpCXpZbZ06zPdBJ0xM/rf/gz0SWJ4oZIThpnEJWWBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b1061287aa1a14595938dcdcc735df3c68479aa91ba559498f55d24405567bdb","last_reissued_at":"2026-07-05T04:06:39.043113Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:06:39.043113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.06417","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":"2026-07-05T04:06:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g41dDQrYjKoazmGUxuSBJY+oK4rRwcJ/lNzn+6S5yGLgQO/KNnlw5QCqQxHPcZKlboG8tRiEhpjJXfI1XJVwBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-14T17:33:29.494523Z"},"content_sha256":"b9bb610cd0d2a311199e8870c5f71c0241f3161eaa61257c6b8bbfcc54a2464d","schema_version":"1.0","event_id":"sha256:b9bb610cd0d2a311199e8870c5f71c0241f3161eaa61257c6b8bbfcc54a2464d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:WEDBFB5KDIKFSWJY3TOMONO7HR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Refined enumeration of vertices among all rooted ordered $d$-trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Heesung Shin, Sangwook Kim, Seunghyun Seo","submitted_at":"2018-06-17T17:34:52Z","abstract_excerpt":"In this paper we enumerate the cardinalities for the set of all vertices of outdegree $\\ge k$ at level $\\ge \\ell$ among all rooted ordered $d$-trees with $n$ edges. Our results unite and generalize several previous works in the literature."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.06417","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/1806.06417/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":"2026-07-05T04:06:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xm7ME7wUPECDb+wcRAUAXUSCyz38seUtDIbZhvGYSSnyDcYAkurmWQVXKQFCRbx/16bRO8Sr3l+Khla8j+M5Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-14T17:33:29.494898Z"},"content_sha256":"5f0cc35831497138a667b36f1cee6307d58bc4bd6237a0d821526498430fcfd6","schema_version":"1.0","event_id":"sha256:5f0cc35831497138a667b36f1cee6307d58bc4bd6237a0d821526498430fcfd6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WEDBFB5KDIKFSWJY3TOMONO7HR/bundle.json","state_url":"https://pith.science/pith/WEDBFB5KDIKFSWJY3TOMONO7HR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WEDBFB5KDIKFSWJY3TOMONO7HR/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":"2026-07-14T17:33:29Z","links":{"resolver":"https://pith.science/pith/WEDBFB5KDIKFSWJY3TOMONO7HR","bundle":"https://pith.science/pith/WEDBFB5KDIKFSWJY3TOMONO7HR/bundle.json","state":"https://pith.science/pith/WEDBFB5KDIKFSWJY3TOMONO7HR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WEDBFB5KDIKFSWJY3TOMONO7HR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:WEDBFB5KDIKFSWJY3TOMONO7HR","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":"28d27df858109503890cbcf9e60819294e9336cb5063f79f7f71f8c408ec8c6c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-17T17:34:52Z","title_canon_sha256":"c7dd4b315f62cc23fe5a3f8101e4079010dbe2d6e2e73452156abd5ef523c208"},"schema_version":"1.0","source":{"id":"1806.06417","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.06417","created_at":"2026-07-05T04:06:39Z"},{"alias_kind":"arxiv_version","alias_value":"1806.06417v1","created_at":"2026-07-05T04:06:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.06417","created_at":"2026-07-05T04:06:39Z"},{"alias_kind":"pith_short_12","alias_value":"WEDBFB5KDIKF","created_at":"2026-07-05T04:06:39Z"},{"alias_kind":"pith_short_16","alias_value":"WEDBFB5KDIKFSWJY","created_at":"2026-07-05T04:06:39Z"},{"alias_kind":"pith_short_8","alias_value":"WEDBFB5K","created_at":"2026-07-05T04:06:39Z"}],"graph_snapshots":[{"event_id":"sha256:5f0cc35831497138a667b36f1cee6307d58bc4bd6237a0d821526498430fcfd6","target":"graph","created_at":"2026-07-05T04:06:39Z","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/1806.06417/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we enumerate the cardinalities for the set of all vertices of outdegree $\\ge k$ at level $\\ge \\ell$ among all rooted ordered $d$-trees with $n$ edges. Our results unite and generalize several previous works in the literature.","authors_text":"Heesung Shin, Sangwook Kim, Seunghyun Seo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-17T17:34:52Z","title":"Refined enumeration of vertices among all rooted ordered $d$-trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.06417","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:b9bb610cd0d2a311199e8870c5f71c0241f3161eaa61257c6b8bbfcc54a2464d","target":"record","created_at":"2026-07-05T04:06:39Z","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":"28d27df858109503890cbcf9e60819294e9336cb5063f79f7f71f8c408ec8c6c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-17T17:34:52Z","title_canon_sha256":"c7dd4b315f62cc23fe5a3f8101e4079010dbe2d6e2e73452156abd5ef523c208"},"schema_version":"1.0","source":{"id":"1806.06417","kind":"arxiv","version":1}},"canonical_sha256":"b1061287aa1a14595938dcdcc735df3c68479aa91ba559498f55d24405567bdb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b1061287aa1a14595938dcdcc735df3c68479aa91ba559498f55d24405567bdb","first_computed_at":"2026-07-05T04:06:39.043113Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:06:39.043113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"R6p9YjM6DHrGV5tUVB9zo69ejMcaEtYG2lt57N9/n7PnOpCXpZbZ06zPdBJ0xM/rf/gz0SWJ4oZIThpnEJWWBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T04:06:39.043463Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.06417","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b9bb610cd0d2a311199e8870c5f71c0241f3161eaa61257c6b8bbfcc54a2464d","sha256:5f0cc35831497138a667b36f1cee6307d58bc4bd6237a0d821526498430fcfd6"],"state_sha256":"f272cda8c3edd23e8a7f62d3ad02a35a8b3d6446faab175ff2189b5701fac61c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4hpZczqk17adsb2ul4ruKzTBysHcsYOdmcHxtWKBSaJI3G6slk4CSydPxozteZ78rCFhhSvyIullMNeg/T4ACA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-14T17:33:29.496912Z","bundle_sha256":"fb0b9ad21d3cd6d0f7814932b0d1adb7deb295f8207dc35f17abbd57fe8001d6"}}