{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:OD5ROD32KONCMJECDBP76LGNZ6","short_pith_number":"pith:OD5ROD32","canonical_record":{"source":{"id":"1809.07465","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-20T04:01:44Z","cross_cats_sorted":[],"title_canon_sha256":"bcb654f41e02f4d4da347d1428dfa3b177af7cfd3bbbc3d6561386a7d1d02a73","abstract_canon_sha256":"da2161900fbe3fa2032e8375c9e548a7b16ac969799ca31f973fa4366e1d0f8e"},"schema_version":"1.0"},"canonical_sha256":"70fb170f7a539a262482185fff2ccdcf86eaa57903201e6b12d6b8f1f23a79b5","source":{"kind":"arxiv","id":"1809.07465","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.07465","created_at":"2026-05-18T00:05:16Z"},{"alias_kind":"arxiv_version","alias_value":"1809.07465v1","created_at":"2026-05-18T00:05:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.07465","created_at":"2026-05-18T00:05:16Z"},{"alias_kind":"pith_short_12","alias_value":"OD5ROD32KONC","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"OD5ROD32KONCMJEC","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"OD5ROD32","created_at":"2026-05-18T12:32:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:OD5ROD32KONCMJECDBP76LGNZ6","target":"record","payload":{"canonical_record":{"source":{"id":"1809.07465","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-20T04:01:44Z","cross_cats_sorted":[],"title_canon_sha256":"bcb654f41e02f4d4da347d1428dfa3b177af7cfd3bbbc3d6561386a7d1d02a73","abstract_canon_sha256":"da2161900fbe3fa2032e8375c9e548a7b16ac969799ca31f973fa4366e1d0f8e"},"schema_version":"1.0"},"canonical_sha256":"70fb170f7a539a262482185fff2ccdcf86eaa57903201e6b12d6b8f1f23a79b5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:16.237300Z","signature_b64":"u644UEGc8MWSyJ7B4+3Qj0WlIIOomBhI5pyvRwEOy331ZOIX5uysf/+quj/VvR2sSlmCKhLm6n82F9DX4Y6xAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70fb170f7a539a262482185fff2ccdcf86eaa57903201e6b12d6b8f1f23a79b5","last_reissued_at":"2026-05-18T00:05:16.236719Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:16.236719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.07465","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":"2026-05-18T00:05:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"am7k1xlliGSwSBolNBG88Uqy5NWGD9QFUNchmdA6z/21OgYmMVG7IxF6dtCSuYuRImT37irFSBdq7rjep6nlBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T02:27:57.366049Z"},"content_sha256":"405039f0d87502626cb9f813459f5cacad32a1edba916cfe5811a46e2a5f9928","schema_version":"1.0","event_id":"sha256:405039f0d87502626cb9f813459f5cacad32a1edba916cfe5811a46e2a5f9928"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:OD5ROD32KONCMJECDBP76LGNZ6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Joint Distributions of Permutation Statistics and the Parabolic Cylinder Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Amy M. Fu, Frank Z.K. Li","submitted_at":"2018-09-20T04:01:44Z","abstract_excerpt":"In this paper, we introduce a context-free grammar $G\\colon x \\rightarrow xy,\\, y \\rightarrow zu,\\,\n  z \\rightarrow zw,\\, w \\rightarrow xv,\\,\n  u \\rightarrow xyz^{-1}v,\\,\n  v \\rightarrow x^{-1}zwu$ over the variable set $V=\\{x,y,z,w,u,v\\}$. We use this grammar to study joint distributions of several permutation statistics related to descents, rises, peaks and valleys. By considering the pattern of an exterior peak, we introduce the exterior peaks of pattern 132 and of pattern 231. Similarly, peaks can also be classified according to their patterns. Let $D$ be the formal derivative operator wit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.07465","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":"2026-05-18T00:05:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m/IREzNf0us+cLueaZrhM0SOiggQH5Eaqjl7K2eLbD/M1LjncezndTkMYpad2nscSAwc13eR0N7HlrtYeZUvAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T02:27:57.366410Z"},"content_sha256":"c5c776f0c7be7fe530bd6b7ae42aef09c2b8251cfb46fb1bf30447c0cf664862","schema_version":"1.0","event_id":"sha256:c5c776f0c7be7fe530bd6b7ae42aef09c2b8251cfb46fb1bf30447c0cf664862"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OD5ROD32KONCMJECDBP76LGNZ6/bundle.json","state_url":"https://pith.science/pith/OD5ROD32KONCMJECDBP76LGNZ6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OD5ROD32KONCMJECDBP76LGNZ6/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":"2026-06-20T02:27:57Z","links":{"resolver":"https://pith.science/pith/OD5ROD32KONCMJECDBP76LGNZ6","bundle":"https://pith.science/pith/OD5ROD32KONCMJECDBP76LGNZ6/bundle.json","state":"https://pith.science/pith/OD5ROD32KONCMJECDBP76LGNZ6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OD5ROD32KONCMJECDBP76LGNZ6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:OD5ROD32KONCMJECDBP76LGNZ6","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":"da2161900fbe3fa2032e8375c9e548a7b16ac969799ca31f973fa4366e1d0f8e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-20T04:01:44Z","title_canon_sha256":"bcb654f41e02f4d4da347d1428dfa3b177af7cfd3bbbc3d6561386a7d1d02a73"},"schema_version":"1.0","source":{"id":"1809.07465","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.07465","created_at":"2026-05-18T00:05:16Z"},{"alias_kind":"arxiv_version","alias_value":"1809.07465v1","created_at":"2026-05-18T00:05:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.07465","created_at":"2026-05-18T00:05:16Z"},{"alias_kind":"pith_short_12","alias_value":"OD5ROD32KONC","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"OD5ROD32KONCMJEC","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"OD5ROD32","created_at":"2026-05-18T12:32:43Z"}],"graph_snapshots":[{"event_id":"sha256:c5c776f0c7be7fe530bd6b7ae42aef09c2b8251cfb46fb1bf30447c0cf664862","target":"graph","created_at":"2026-05-18T00:05:16Z","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"},"paper":{"abstract_excerpt":"In this paper, we introduce a context-free grammar $G\\colon x \\rightarrow xy,\\, y \\rightarrow zu,\\,\n  z \\rightarrow zw,\\, w \\rightarrow xv,\\,\n  u \\rightarrow xyz^{-1}v,\\,\n  v \\rightarrow x^{-1}zwu$ over the variable set $V=\\{x,y,z,w,u,v\\}$. We use this grammar to study joint distributions of several permutation statistics related to descents, rises, peaks and valleys. By considering the pattern of an exterior peak, we introduce the exterior peaks of pattern 132 and of pattern 231. Similarly, peaks can also be classified according to their patterns. Let $D$ be the formal derivative operator wit","authors_text":"Amy M. Fu, Frank Z.K. Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-20T04:01:44Z","title":"Joint Distributions of Permutation Statistics and the Parabolic Cylinder Functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.07465","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":"sha256:405039f0d87502626cb9f813459f5cacad32a1edba916cfe5811a46e2a5f9928","target":"record","created_at":"2026-05-18T00:05:16Z","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":"da2161900fbe3fa2032e8375c9e548a7b16ac969799ca31f973fa4366e1d0f8e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-20T04:01:44Z","title_canon_sha256":"bcb654f41e02f4d4da347d1428dfa3b177af7cfd3bbbc3d6561386a7d1d02a73"},"schema_version":"1.0","source":{"id":"1809.07465","kind":"arxiv","version":1}},"canonical_sha256":"70fb170f7a539a262482185fff2ccdcf86eaa57903201e6b12d6b8f1f23a79b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"70fb170f7a539a262482185fff2ccdcf86eaa57903201e6b12d6b8f1f23a79b5","first_computed_at":"2026-05-18T00:05:16.236719Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:05:16.236719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u644UEGc8MWSyJ7B4+3Qj0WlIIOomBhI5pyvRwEOy331ZOIX5uysf/+quj/VvR2sSlmCKhLm6n82F9DX4Y6xAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:05:16.237300Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.07465","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:405039f0d87502626cb9f813459f5cacad32a1edba916cfe5811a46e2a5f9928","sha256:c5c776f0c7be7fe530bd6b7ae42aef09c2b8251cfb46fb1bf30447c0cf664862"],"state_sha256":"5052cfafcf7336f2aff7d3b476058cd9f1fdd197d79cb16c8d5aae6922cf4759"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M/kJ7hb78JynbuETTCwlJuV0qP9ZUPZlIS7/vQCyrIkjcOn4x7WV6GLEwfi0LA77U29PDlx01ApnaMQyczI+DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T02:27:57.368408Z","bundle_sha256":"61f9ed81bfea4f47b5891343b57f5423acb773cb6faba8b4c4404f03200e1da5"}}