{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:54WIHGEUJXQBZ5MEAJ275UTOPR","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":"f0907dbdcd461ce0d50502f72243c749cbc8f5ce86e341af6a488572dc80f7ca","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-12-21T13:52:16Z","title_canon_sha256":"56f6556a4912cfdb2098ff23208d7c34ea4c863a90a14c9e91a0c224c726067f"},"schema_version":"1.0","source":{"id":"1012.4654","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.4654","created_at":"2026-05-18T04:30:25Z"},{"alias_kind":"arxiv_version","alias_value":"1012.4654v2","created_at":"2026-05-18T04:30:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.4654","created_at":"2026-05-18T04:30:25Z"},{"alias_kind":"pith_short_12","alias_value":"54WIHGEUJXQB","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"54WIHGEUJXQBZ5ME","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"54WIHGEU","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:5bcbaa89cd965c0956f85c56fddf29f7856797e6b64605d1177d53947575c877","target":"graph","created_at":"2026-05-18T04:30:25Z","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 provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem.\n  Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more standard approach to Cayley's formula. Moreover, this algebraic structure sheds light on the problem of counting the unlabeled trees. In particular, it indicates how counting the number of unlabeled trees on $n$ vertices is connected to finding the number of partitions of $n-2$","authors_text":"Giulio Iacobelli, Victor N. Ermolaev","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-12-21T13:52:16Z","title":"A note on counting labeled and unlabeled trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.4654","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:d935491bbc85a2a51c164212cd52144483673fad9288ebce1c865ada3d850e6f","target":"record","created_at":"2026-05-18T04:30:25Z","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":"f0907dbdcd461ce0d50502f72243c749cbc8f5ce86e341af6a488572dc80f7ca","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-12-21T13:52:16Z","title_canon_sha256":"56f6556a4912cfdb2098ff23208d7c34ea4c863a90a14c9e91a0c224c726067f"},"schema_version":"1.0","source":{"id":"1012.4654","kind":"arxiv","version":2}},"canonical_sha256":"ef2c8398944de01cf5840275fed26e7c557cafe09515787f41f05e46f686a97b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ef2c8398944de01cf5840275fed26e7c557cafe09515787f41f05e46f686a97b","first_computed_at":"2026-05-18T04:30:25.338532Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:30:25.338532Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LYFWZaH7A1imKzPUAGIeXf1a+P68pdV8xBjXGROB5ySCQLWDK6rT4iyXOUvDjte3pmLvNlWcACSclUNb+SP0DA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:30:25.339014Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.4654","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d935491bbc85a2a51c164212cd52144483673fad9288ebce1c865ada3d850e6f","sha256:5bcbaa89cd965c0956f85c56fddf29f7856797e6b64605d1177d53947575c877"],"state_sha256":"5783a6abac4b7b698c7854d0fe4d421094eaf3fd0c8241d7bf31ffd1795bf5f6"}