{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:TVT7KHXBYFG4SVKTIBUN3MHWAU","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":"a27c0720e83a62df72de48a5df729fbfb2fdafd37c3ce6f273948deb1626af6c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-08-30T09:46:07Z","title_canon_sha256":"82d6e788cacd8397a71f3331c93029e1eacc44753284a440bb9834c1bca8e8a1"},"schema_version":"1.0","source":{"id":"2509.00436","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.00436","created_at":"2026-07-05T12:02:16Z"},{"alias_kind":"arxiv_version","alias_value":"2509.00436v1","created_at":"2026-07-05T12:02:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.00436","created_at":"2026-07-05T12:02:16Z"},{"alias_kind":"pith_short_12","alias_value":"TVT7KHXBYFG4","created_at":"2026-07-05T12:02:16Z"},{"alias_kind":"pith_short_16","alias_value":"TVT7KHXBYFG4SVKT","created_at":"2026-07-05T12:02:16Z"},{"alias_kind":"pith_short_8","alias_value":"TVT7KHXB","created_at":"2026-07-05T12:02:16Z"}],"graph_snapshots":[{"event_id":"sha256:56407e635c88778e64bc7a394459dea7ffc773803860fb76a7792bd2b820cdcc","target":"graph","created_at":"2026-07-05T12:02:16Z","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/2509.00436/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we explore parking distributions on caterpillar trees, focusing on two primary statistics: the number of lucky cars and the frequency with which cars prefer specific parking spaces. We use first-return decomposition to reveal a symmetry in their joint distribution and develop a $q, t$-analog of the Fuss-Catalan generating function. We prove that this generating function exhibits specific symmetry and satisfies a functional equation. Additionally, we extend our findings to any m statistics that satisfy certain criteria, presenting a concrete example of such $m$ statistics to illu","authors_text":"Amanuel T. Getachew","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-08-30T09:46:07Z","title":"Symmetry in Tree Parking Distributions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.00436","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":"sha256:7e4f2c18615b9af87e658e379d2c32201608aef33e5ad19a1a75321b862d942f","target":"record","created_at":"2026-07-05T12:02:16Z","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":"a27c0720e83a62df72de48a5df729fbfb2fdafd37c3ce6f273948deb1626af6c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-08-30T09:46:07Z","title_canon_sha256":"82d6e788cacd8397a71f3331c93029e1eacc44753284a440bb9834c1bca8e8a1"},"schema_version":"1.0","source":{"id":"2509.00436","kind":"arxiv","version":1}},"canonical_sha256":"9d67f51ee1c14dc955534068ddb0f60522762e0dd2d7273832badae0c205fcaa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9d67f51ee1c14dc955534068ddb0f60522762e0dd2d7273832badae0c205fcaa","first_computed_at":"2026-07-05T12:02:16.320074Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:02:16.320074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h5KdMjLyc1t4QLDqHlv1bXh9FJ0iU2xRg7pJwVkrym4i1BdzOW/nHwQk76wGXVPgm2Z+91i+VM58C14tVq2LBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T12:02:16.320608Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.00436","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7e4f2c18615b9af87e658e379d2c32201608aef33e5ad19a1a75321b862d942f","sha256:56407e635c88778e64bc7a394459dea7ffc773803860fb76a7792bd2b820cdcc"],"state_sha256":"ba2b14551affa328f50761967348156cef3b48356f19d8f9e32bf480036deecc"}