{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:KLZ6IHP4K6UDFFF7TBIBYNECI3","short_pith_number":"pith:KLZ6IHP4","canonical_record":{"source":{"id":"1501.07468","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-29T14:54:18Z","cross_cats_sorted":[],"title_canon_sha256":"3d22e0bad1671ff65f4f81c090b0ce4c23af674c17e6703905aa4e5b746f47f4","abstract_canon_sha256":"29807d5634723a4647a694dc34db89fdeec4640cce2ef521719ec6f21a07a8a4"},"schema_version":"1.0"},"canonical_sha256":"52f3e41dfc57a83294bf98501c348246ce701a6a1e0b066041b0982436bc6f89","source":{"kind":"arxiv","id":"1501.07468","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.07468","created_at":"2026-05-17T23:51:06Z"},{"alias_kind":"arxiv_version","alias_value":"1501.07468v3","created_at":"2026-05-17T23:51:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.07468","created_at":"2026-05-17T23:51:06Z"},{"alias_kind":"pith_short_12","alias_value":"KLZ6IHP4K6UD","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"KLZ6IHP4K6UDFFF7","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"KLZ6IHP4","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:KLZ6IHP4K6UDFFF7TBIBYNECI3","target":"record","payload":{"canonical_record":{"source":{"id":"1501.07468","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-29T14:54:18Z","cross_cats_sorted":[],"title_canon_sha256":"3d22e0bad1671ff65f4f81c090b0ce4c23af674c17e6703905aa4e5b746f47f4","abstract_canon_sha256":"29807d5634723a4647a694dc34db89fdeec4640cce2ef521719ec6f21a07a8a4"},"schema_version":"1.0"},"canonical_sha256":"52f3e41dfc57a83294bf98501c348246ce701a6a1e0b066041b0982436bc6f89","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:06.530841Z","signature_b64":"mYFjGN9FeHBBZVeCYVUeNq6yUPUhv/cFh4egsDFcSP+BZ4l9Ktyv6ZmToO5w8e/7mgiPY3kYA+fqUhD+xxzfDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52f3e41dfc57a83294bf98501c348246ce701a6a1e0b066041b0982436bc6f89","last_reissued_at":"2026-05-17T23:51:06.530200Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:06.530200Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.07468","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:51:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"07looBfdOJcOXM8RiAOP5J9HEbTySxyKnOwXml76X/XIMVY7UvJtTJd2WyBNBQ4S4S5etAdb35NKZRNmyCVJCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T09:23:44.317506Z"},"content_sha256":"7145c3a977977275c6b6fae45fa22c1e395b66c0dfa7ec83cb90cce4c8d967c6","schema_version":"1.0","event_id":"sha256:7145c3a977977275c6b6fae45fa22c1e395b66c0dfa7ec83cb90cce4c8d967c6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:KLZ6IHP4K6UDFFF7TBIBYNECI3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting vertices in plane and $k$-ary trees with given outdegree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jia He, Rosena R. X. Du, Xueli Yun","submitted_at":"2015-01-29T14:54:18Z","abstract_excerpt":"We count the number of vertices in plane trees and $k$-ary trees with given outdegree, and prove that the total number of vertices of outdegree $i$ over all plane trees with $n$ edges is ${2n-i-1 \\choose n-1}$, and the total number of vertices of outdegree $i$ over all $k$-ary trees with $n$ edges is ${k\\choose i}{kn\\choose n-i}$. For both results we give bijective proofs as well as generating function proofs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.07468","kind":"arxiv","version":3},"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":"2026-05-17T23:51:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7GcTSCQzmp9Dh0Fuoc+7KluVwKCAi4nEs85aMEey12rBgh+O8Ugt5Zjv/+Q65XXNprKmfghNUJM/BMS5SO5aAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T09:23:44.317847Z"},"content_sha256":"7e4ebb6a351fa8afbedf7986f971010b2452ae83f8840ff3f75665a427f8e869","schema_version":"1.0","event_id":"sha256:7e4ebb6a351fa8afbedf7986f971010b2452ae83f8840ff3f75665a427f8e869"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KLZ6IHP4K6UDFFF7TBIBYNECI3/bundle.json","state_url":"https://pith.science/pith/KLZ6IHP4K6UDFFF7TBIBYNECI3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KLZ6IHP4K6UDFFF7TBIBYNECI3/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":"2026-06-03T09:23:44Z","links":{"resolver":"https://pith.science/pith/KLZ6IHP4K6UDFFF7TBIBYNECI3","bundle":"https://pith.science/pith/KLZ6IHP4K6UDFFF7TBIBYNECI3/bundle.json","state":"https://pith.science/pith/KLZ6IHP4K6UDFFF7TBIBYNECI3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KLZ6IHP4K6UDFFF7TBIBYNECI3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:KLZ6IHP4K6UDFFF7TBIBYNECI3","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":"29807d5634723a4647a694dc34db89fdeec4640cce2ef521719ec6f21a07a8a4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-29T14:54:18Z","title_canon_sha256":"3d22e0bad1671ff65f4f81c090b0ce4c23af674c17e6703905aa4e5b746f47f4"},"schema_version":"1.0","source":{"id":"1501.07468","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.07468","created_at":"2026-05-17T23:51:06Z"},{"alias_kind":"arxiv_version","alias_value":"1501.07468v3","created_at":"2026-05-17T23:51:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.07468","created_at":"2026-05-17T23:51:06Z"},{"alias_kind":"pith_short_12","alias_value":"KLZ6IHP4K6UD","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"KLZ6IHP4K6UDFFF7","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"KLZ6IHP4","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:7e4ebb6a351fa8afbedf7986f971010b2452ae83f8840ff3f75665a427f8e869","target":"graph","created_at":"2026-05-17T23:51:06Z","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":"We count the number of vertices in plane trees and $k$-ary trees with given outdegree, and prove that the total number of vertices of outdegree $i$ over all plane trees with $n$ edges is ${2n-i-1 \\choose n-1}$, and the total number of vertices of outdegree $i$ over all $k$-ary trees with $n$ edges is ${k\\choose i}{kn\\choose n-i}$. For both results we give bijective proofs as well as generating function proofs.","authors_text":"Jia He, Rosena R. X. Du, Xueli Yun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-29T14:54:18Z","title":"Counting vertices in plane and $k$-ary trees with given outdegree"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.07468","kind":"arxiv","version":3},"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":"sha256:7145c3a977977275c6b6fae45fa22c1e395b66c0dfa7ec83cb90cce4c8d967c6","target":"record","created_at":"2026-05-17T23:51:06Z","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":"29807d5634723a4647a694dc34db89fdeec4640cce2ef521719ec6f21a07a8a4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-29T14:54:18Z","title_canon_sha256":"3d22e0bad1671ff65f4f81c090b0ce4c23af674c17e6703905aa4e5b746f47f4"},"schema_version":"1.0","source":{"id":"1501.07468","kind":"arxiv","version":3}},"canonical_sha256":"52f3e41dfc57a83294bf98501c348246ce701a6a1e0b066041b0982436bc6f89","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"52f3e41dfc57a83294bf98501c348246ce701a6a1e0b066041b0982436bc6f89","first_computed_at":"2026-05-17T23:51:06.530200Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:06.530200Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mYFjGN9FeHBBZVeCYVUeNq6yUPUhv/cFh4egsDFcSP+BZ4l9Ktyv6ZmToO5w8e/7mgiPY3kYA+fqUhD+xxzfDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:06.530841Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.07468","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7145c3a977977275c6b6fae45fa22c1e395b66c0dfa7ec83cb90cce4c8d967c6","sha256:7e4ebb6a351fa8afbedf7986f971010b2452ae83f8840ff3f75665a427f8e869"],"state_sha256":"e0361ac4808ced61ea48d0468c294f478ebebd46656fd5ad36051d3fdfdd6fa4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3mQb2o9PgiNI7JvdT6ssQUeBwQVFwPwIFjeQ46HJVgNMCdrtXhZ8DEWACiP9RmzaT+Vc+QvVVaL/sXWY2qM5Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T09:23:44.319743Z","bundle_sha256":"c85c9e39df0387352ec745ad40f75994c1597cf842eb90a57ff067395a6df535"}}