{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:FVZZIF74GLXRIRV2RB7TKG7KGC","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":"b587fe164aa3518e0d657b50fe16cc2018722978c9915e3cb94c036285c05fa2","cross_cats_sorted":["math.GT","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-18T22:39:50Z","title_canon_sha256":"4b3e36bc805ef9a45c1670ba2677ba26d4e58a25cc76a7d1d30e26c9dcb18947"},"schema_version":"1.0","source":{"id":"1704.05557","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.05557","created_at":"2026-05-18T00:46:07Z"},{"alias_kind":"arxiv_version","alias_value":"1704.05557v1","created_at":"2026-05-18T00:46:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.05557","created_at":"2026-05-18T00:46:07Z"},{"alias_kind":"pith_short_12","alias_value":"FVZZIF74GLXR","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"FVZZIF74GLXRIRV2","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"FVZZIF74","created_at":"2026-05-18T12:31:15Z"}],"graph_snapshots":[{"event_id":"sha256:a4d7ef91ecf597dc9627fa0dbb085cc72fbad2261c2776c0e85ca750e8db63d5","target":"graph","created_at":"2026-05-18T00:46:07Z","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 give a geometric realization of the polyhedra governed by the structure of associative algebras with co-inner products, or more precisely, governed by directed planar trees. Our explicit realization of these polyhedra, which include the associahedra in a special case, shows in particular that these polyhedra are homeomorphic to balls. We also calculate the number of vertices of the lowest generalized associahedra, giving appropriate generalizations of the Catalan numbers.","authors_text":"Kate Poirier, Thomas Tradler","cross_cats":["math.GT","math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-18T22:39:50Z","title":"The Combinatorics of Directed Planar Trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.05557","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":"sha256:01d064540ea0103ca0f35dc019243245a9f6254dc50fe0dc12d794677b78be30","target":"record","created_at":"2026-05-18T00:46:07Z","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":"b587fe164aa3518e0d657b50fe16cc2018722978c9915e3cb94c036285c05fa2","cross_cats_sorted":["math.GT","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-18T22:39:50Z","title_canon_sha256":"4b3e36bc805ef9a45c1670ba2677ba26d4e58a25cc76a7d1d30e26c9dcb18947"},"schema_version":"1.0","source":{"id":"1704.05557","kind":"arxiv","version":1}},"canonical_sha256":"2d739417fc32ef1446ba887f351bea309fd8374bbaf015215484d6f784dfa606","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2d739417fc32ef1446ba887f351bea309fd8374bbaf015215484d6f784dfa606","first_computed_at":"2026-05-18T00:46:07.140783Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:46:07.140783Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JY6wCxA/vsa2q5RvGEiGyEluseNuP8m3wcg1xwkzR7tqV2B2krBggvQu2/PFqiZ/12tIv6YXBYSTAvjaSP5MDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:46:07.141389Z","signed_message":"canonical_sha256_bytes"},"source_id":"1704.05557","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:01d064540ea0103ca0f35dc019243245a9f6254dc50fe0dc12d794677b78be30","sha256:a4d7ef91ecf597dc9627fa0dbb085cc72fbad2261c2776c0e85ca750e8db63d5"],"state_sha256":"a704aca7ff03a9d97bd3b06a5adefdfd65d30f29529109e83686b633c1863af6"}