{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:CMJPHLZ4W3KYXJ3XFCOH6TH45Z","short_pith_number":"pith:CMJPHLZ4","canonical_record":{"source":{"id":"1205.0369","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-02T10:22:03Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"e3f6e2b292085ebfbd482f481b08d16d0a1af2c0fc5fb15d96fd141fe5df8d6c","abstract_canon_sha256":"9db324e25033d7810176bcb7039a58e7828e387a08af3138286ed9131e3d1b5f"},"schema_version":"1.0"},"canonical_sha256":"1312f3af3cb6d58ba777289c7f4cfcee6cf3a31df732788a8c40259a6e447f6a","source":{"kind":"arxiv","id":"1205.0369","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.0369","created_at":"2026-05-18T03:27:34Z"},{"alias_kind":"arxiv_version","alias_value":"1205.0369v2","created_at":"2026-05-18T03:27:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.0369","created_at":"2026-05-18T03:27:34Z"},{"alias_kind":"pith_short_12","alias_value":"CMJPHLZ4W3KY","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CMJPHLZ4W3KYXJ3X","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CMJPHLZ4","created_at":"2026-05-18T12:27:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:CMJPHLZ4W3KYXJ3XFCOH6TH45Z","target":"record","payload":{"canonical_record":{"source":{"id":"1205.0369","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-02T10:22:03Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"e3f6e2b292085ebfbd482f481b08d16d0a1af2c0fc5fb15d96fd141fe5df8d6c","abstract_canon_sha256":"9db324e25033d7810176bcb7039a58e7828e387a08af3138286ed9131e3d1b5f"},"schema_version":"1.0"},"canonical_sha256":"1312f3af3cb6d58ba777289c7f4cfcee6cf3a31df732788a8c40259a6e447f6a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:27:34.467514Z","signature_b64":"9KxbhojjLLeNMLmVflTzl7jrBUDxhOuNzLHgmOie0nmBrpiVnew87ly72zWirzPf2D+auhG8yt/s1H+M1773DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1312f3af3cb6d58ba777289c7f4cfcee6cf3a31df732788a8c40259a6e447f6a","last_reissued_at":"2026-05-18T03:27:34.467031Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:27:34.467031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1205.0369","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":"2026-05-18T03:27:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mzuT8vRdiyIWz4EpcZaOCuMOqQgTcl8rov2f87F+D4iXl7U+IW25kohVazbZjdinyNFmDT4t5BkNnmhCVRYbDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:28:43.365413Z"},"content_sha256":"a6048e132ac31974587e01a104e1b06cfc63924b5db24bc978e72de27bb665f5","schema_version":"1.0","event_id":"sha256:a6048e132ac31974587e01a104e1b06cfc63924b5db24bc978e72de27bb665f5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:CMJPHLZ4W3KYXJ3XFCOH6TH45Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A multivariate hook formula for labelled trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"I. P. Goulden, Valentin F\\'eray","submitted_at":"2012-05-02T10:22:03Z","abstract_excerpt":"Several hook summation formulae for binary trees have appeared recently in the literature. In this paper we present an analogous formula for unordered increasing trees of size r, which involves r parameters. The right-hand side can be written nicely as a product of linear factors. We study two specializations of this new formula, including Cayley's enumeration of trees with respect to vertex degree. We give three proofs of the hook formula. One of these proofs arises somewhat indirectly, from representation theory of the symmetric groups, and in particular uses Kerov's character polynomials. T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.0369","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":"2026-05-18T03:27:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VkqhupPPU5gTc/+NKCzyUsmsAacNGP+pNVuKQRjD7TXOTlAPkQEiUXFnpk7NqWrwAas6Cl9Kx5icSs51/5c2DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:28:43.365786Z"},"content_sha256":"97eaa4346d3b0849364f0b0eeb7343ecec512dd2399f7edd118371005de0f327","schema_version":"1.0","event_id":"sha256:97eaa4346d3b0849364f0b0eeb7343ecec512dd2399f7edd118371005de0f327"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CMJPHLZ4W3KYXJ3XFCOH6TH45Z/bundle.json","state_url":"https://pith.science/pith/CMJPHLZ4W3KYXJ3XFCOH6TH45Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CMJPHLZ4W3KYXJ3XFCOH6TH45Z/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":"2026-06-23T16:28:43Z","links":{"resolver":"https://pith.science/pith/CMJPHLZ4W3KYXJ3XFCOH6TH45Z","bundle":"https://pith.science/pith/CMJPHLZ4W3KYXJ3XFCOH6TH45Z/bundle.json","state":"https://pith.science/pith/CMJPHLZ4W3KYXJ3XFCOH6TH45Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CMJPHLZ4W3KYXJ3XFCOH6TH45Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:CMJPHLZ4W3KYXJ3XFCOH6TH45Z","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":"9db324e25033d7810176bcb7039a58e7828e387a08af3138286ed9131e3d1b5f","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-02T10:22:03Z","title_canon_sha256":"e3f6e2b292085ebfbd482f481b08d16d0a1af2c0fc5fb15d96fd141fe5df8d6c"},"schema_version":"1.0","source":{"id":"1205.0369","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.0369","created_at":"2026-05-18T03:27:34Z"},{"alias_kind":"arxiv_version","alias_value":"1205.0369v2","created_at":"2026-05-18T03:27:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.0369","created_at":"2026-05-18T03:27:34Z"},{"alias_kind":"pith_short_12","alias_value":"CMJPHLZ4W3KY","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CMJPHLZ4W3KYXJ3X","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CMJPHLZ4","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:97eaa4346d3b0849364f0b0eeb7343ecec512dd2399f7edd118371005de0f327","target":"graph","created_at":"2026-05-18T03:27:34Z","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":"Several hook summation formulae for binary trees have appeared recently in the literature. In this paper we present an analogous formula for unordered increasing trees of size r, which involves r parameters. The right-hand side can be written nicely as a product of linear factors. We study two specializations of this new formula, including Cayley's enumeration of trees with respect to vertex degree. We give three proofs of the hook formula. One of these proofs arises somewhat indirectly, from representation theory of the symmetric groups, and in particular uses Kerov's character polynomials. T","authors_text":"I. P. Goulden, Valentin F\\'eray","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-02T10:22:03Z","title":"A multivariate hook formula for labelled trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.0369","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:a6048e132ac31974587e01a104e1b06cfc63924b5db24bc978e72de27bb665f5","target":"record","created_at":"2026-05-18T03:27:34Z","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":"9db324e25033d7810176bcb7039a58e7828e387a08af3138286ed9131e3d1b5f","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-02T10:22:03Z","title_canon_sha256":"e3f6e2b292085ebfbd482f481b08d16d0a1af2c0fc5fb15d96fd141fe5df8d6c"},"schema_version":"1.0","source":{"id":"1205.0369","kind":"arxiv","version":2}},"canonical_sha256":"1312f3af3cb6d58ba777289c7f4cfcee6cf3a31df732788a8c40259a6e447f6a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1312f3af3cb6d58ba777289c7f4cfcee6cf3a31df732788a8c40259a6e447f6a","first_computed_at":"2026-05-18T03:27:34.467031Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:27:34.467031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9KxbhojjLLeNMLmVflTzl7jrBUDxhOuNzLHgmOie0nmBrpiVnew87ly72zWirzPf2D+auhG8yt/s1H+M1773DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:27:34.467514Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.0369","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a6048e132ac31974587e01a104e1b06cfc63924b5db24bc978e72de27bb665f5","sha256:97eaa4346d3b0849364f0b0eeb7343ecec512dd2399f7edd118371005de0f327"],"state_sha256":"de3eaa230b8d9e16189dfcbbf2404aac1e7253f8caa2e666bd469f635437e9cf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tLFCoefFUto6RDE3Ik+gXHnaPoSDqaeU7Rcd0K/SuumjsVzU5n4woQCsuEcbUMgfEN0qstz/qmSQeRzbthNsDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T16:28:43.367825Z","bundle_sha256":"240af882bec516806b7f2b37b44bca4cd5e9aefb97d5c2a54a0406f289b239c3"}}