{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:3IG5TVMYWRQZCPBBB3FFJOKVCB","short_pith_number":"pith:3IG5TVMY","canonical_record":{"source":{"id":"1904.03519","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2019-04-06T19:58:27Z","cross_cats_sorted":[],"title_canon_sha256":"835902952d2865a03f855739f689c7084eafddb5342d49a3d68c78cb5e5511a2","abstract_canon_sha256":"ad7fa5cb3808c42d5b166cef021891db61acc8b4b84292be80e2c5c212b39417"},"schema_version":"1.0"},"canonical_sha256":"da0dd9d598b461913c210eca54b955104b40fdd9f8e7f4c94f89f023a987ed57","source":{"kind":"arxiv","id":"1904.03519","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.03519","created_at":"2026-05-17T23:49:13Z"},{"alias_kind":"arxiv_version","alias_value":"1904.03519v1","created_at":"2026-05-17T23:49:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.03519","created_at":"2026-05-17T23:49:13Z"},{"alias_kind":"pith_short_12","alias_value":"3IG5TVMYWRQZ","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"3IG5TVMYWRQZCPBB","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"3IG5TVMY","created_at":"2026-05-18T12:33:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:3IG5TVMYWRQZCPBBB3FFJOKVCB","target":"record","payload":{"canonical_record":{"source":{"id":"1904.03519","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2019-04-06T19:58:27Z","cross_cats_sorted":[],"title_canon_sha256":"835902952d2865a03f855739f689c7084eafddb5342d49a3d68c78cb5e5511a2","abstract_canon_sha256":"ad7fa5cb3808c42d5b166cef021891db61acc8b4b84292be80e2c5c212b39417"},"schema_version":"1.0"},"canonical_sha256":"da0dd9d598b461913c210eca54b955104b40fdd9f8e7f4c94f89f023a987ed57","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:13.115274Z","signature_b64":"PoGyoBYo/hPWEWNqBiE4gh1kT0E8vSjdE5rwQ2PtKlFiZW9slHRk6XC8ni4hQWa8zSSMACME0fBijbXqs6ilAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da0dd9d598b461913c210eca54b955104b40fdd9f8e7f4c94f89f023a987ed57","last_reissued_at":"2026-05-17T23:49:13.114496Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:13.114496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.03519","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":"2026-05-17T23:49:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TN6m5qx9ct1/unirpt5g6hUxsklN1CoFhDMnU3VVyqlYALS+R4sbZb++1I/59ABXFnlxflZJznA1oInkjN9dBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T06:59:49.931073Z"},"content_sha256":"a8be99a01f5be4cffa94604cbebd0e214a5053a2de4756910262cd883e512e07","schema_version":"1.0","event_id":"sha256:a8be99a01f5be4cffa94604cbebd0e214a5053a2de4756910262cd883e512e07"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:3IG5TVMYWRQZCPBBB3FFJOKVCB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Protection numbers in simply generated trees and P\\'olya trees","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bernhard Gittenberger, Isabella Larcher, Ma{\\l}gorzata Sulkowska, Zbigniew Go{\\l}\\k{e}biewski","submitted_at":"2019-04-06T19:58:27Z","abstract_excerpt":"We determine the limit of the expected value and the variance of the protection number of the root in simply generated trees, in P\\'olya trees, and in unlabelled non-plane binary trees, when the number of vertices tends to infinity. Moreover, we compute expectation and variance of the protection number of a randomly chosen vertex in all those tree classes. We obtain exact formulas as sum representations, where the obtained sums are rapidly converging and therefore allowing an efficient numerical computation of high accuracy. Most proofs are based on a singularity analysis of generating functio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.03519","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":""},"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":"2026-05-17T23:49:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YvXKvEdb4laU8CFXzBC0SZeipQwYGhiNrCPmexf9b1Od1/qqulBpT1OE5TVyLUbQV0I3pgR6p7eH5kr+CD8dDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T06:59:49.931748Z"},"content_sha256":"b680b4e325caf1236fe436ce3bea489984cddbe30f30bc63c1b6b13c915f43d5","schema_version":"1.0","event_id":"sha256:b680b4e325caf1236fe436ce3bea489984cddbe30f30bc63c1b6b13c915f43d5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3IG5TVMYWRQZCPBBB3FFJOKVCB/bundle.json","state_url":"https://pith.science/pith/3IG5TVMYWRQZCPBBB3FFJOKVCB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3IG5TVMYWRQZCPBBB3FFJOKVCB/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":"2026-05-27T06:59:49Z","links":{"resolver":"https://pith.science/pith/3IG5TVMYWRQZCPBBB3FFJOKVCB","bundle":"https://pith.science/pith/3IG5TVMYWRQZCPBBB3FFJOKVCB/bundle.json","state":"https://pith.science/pith/3IG5TVMYWRQZCPBBB3FFJOKVCB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3IG5TVMYWRQZCPBBB3FFJOKVCB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:3IG5TVMYWRQZCPBBB3FFJOKVCB","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":"ad7fa5cb3808c42d5b166cef021891db61acc8b4b84292be80e2c5c212b39417","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2019-04-06T19:58:27Z","title_canon_sha256":"835902952d2865a03f855739f689c7084eafddb5342d49a3d68c78cb5e5511a2"},"schema_version":"1.0","source":{"id":"1904.03519","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.03519","created_at":"2026-05-17T23:49:13Z"},{"alias_kind":"arxiv_version","alias_value":"1904.03519v1","created_at":"2026-05-17T23:49:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.03519","created_at":"2026-05-17T23:49:13Z"},{"alias_kind":"pith_short_12","alias_value":"3IG5TVMYWRQZ","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"3IG5TVMYWRQZCPBB","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"3IG5TVMY","created_at":"2026-05-18T12:33:07Z"}],"graph_snapshots":[{"event_id":"sha256:b680b4e325caf1236fe436ce3bea489984cddbe30f30bc63c1b6b13c915f43d5","target":"graph","created_at":"2026-05-17T23:49:13Z","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 determine the limit of the expected value and the variance of the protection number of the root in simply generated trees, in P\\'olya trees, and in unlabelled non-plane binary trees, when the number of vertices tends to infinity. Moreover, we compute expectation and variance of the protection number of a randomly chosen vertex in all those tree classes. We obtain exact formulas as sum representations, where the obtained sums are rapidly converging and therefore allowing an efficient numerical computation of high accuracy. Most proofs are based on a singularity analysis of generating functio","authors_text":"Bernhard Gittenberger, Isabella Larcher, Ma{\\l}gorzata Sulkowska, Zbigniew Go{\\l}\\k{e}biewski","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2019-04-06T19:58:27Z","title":"Protection numbers in simply generated trees and P\\'olya trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.03519","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:a8be99a01f5be4cffa94604cbebd0e214a5053a2de4756910262cd883e512e07","target":"record","created_at":"2026-05-17T23:49:13Z","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":"ad7fa5cb3808c42d5b166cef021891db61acc8b4b84292be80e2c5c212b39417","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2019-04-06T19:58:27Z","title_canon_sha256":"835902952d2865a03f855739f689c7084eafddb5342d49a3d68c78cb5e5511a2"},"schema_version":"1.0","source":{"id":"1904.03519","kind":"arxiv","version":1}},"canonical_sha256":"da0dd9d598b461913c210eca54b955104b40fdd9f8e7f4c94f89f023a987ed57","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"da0dd9d598b461913c210eca54b955104b40fdd9f8e7f4c94f89f023a987ed57","first_computed_at":"2026-05-17T23:49:13.114496Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:49:13.114496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PoGyoBYo/hPWEWNqBiE4gh1kT0E8vSjdE5rwQ2PtKlFiZW9slHRk6XC8ni4hQWa8zSSMACME0fBijbXqs6ilAA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:49:13.115274Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.03519","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a8be99a01f5be4cffa94604cbebd0e214a5053a2de4756910262cd883e512e07","sha256:b680b4e325caf1236fe436ce3bea489984cddbe30f30bc63c1b6b13c915f43d5"],"state_sha256":"8d86dbb3156e3dcdd32c2e29cc42959fc30aff4783a39ddc4303be3e0077198d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uIvvhmYC2Sw/unBX6epXP4HzY0t4i1+3o8MqBiseDms01UAN5N+qSAmkvlMDRyZWvY/VUC0opqUmwlCg1RQCBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T06:59:49.935247Z","bundle_sha256":"0403e96599a9abd66288622b05d54cf8395c15f64090d494a1ddd97497e8c2cf"}}