{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:IBQCV5BHN2GR72PNETUFMWSD5T","short_pith_number":"pith:IBQCV5BH","schema_version":"1.0","canonical_sha256":"40602af4276e8d1fe9ed24e8565a43ece57210fc8565150a2c009076c9336ebd","source":{"kind":"arxiv","id":"1405.5587","version":2},"attestation_state":"computed","paper":{"title":"Parking functions, Shi arrangements, and mixed graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Amanda Ruiz, Ana Berrizbeitia, Claudia Rodriguez, Matthias Beck, Michael Dairyko, Schuyler Veeneman","submitted_at":"2014-05-22T01:22:01Z","abstract_excerpt":"The \\emph{Shi arrangement} is the set of all hyperplanes in $\\mathbb R^n$ of the form $x_j - x_k = 0$ or $1$ for $1 \\le j < k \\le n$. Shi observed in 1986 that the number of regions (i.e., connected components of the complement) of this arrangement is $(n+1)^{n-1}$. An unrelated combinatorial concept is that of a \\emph{parking function}, i.e., a sequence $(x_1, x_2, ..., x_n)$ of positive integers that, when rearranged from smallest to largest, satisfies $x_k \\le k$. (There is an illustrative reason for the term \\emph{parking function}.) It turns out that the number of parking functions of len"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1405.5587","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-22T01:22:01Z","cross_cats_sorted":[],"title_canon_sha256":"cfa482244c535393e70a5aaa11ea880e28ea13a07d0a611a9967ffad3e47a084","abstract_canon_sha256":"e403b1453ac87ac07edd1baee326e8298ccf4d26e4d89095175e6d6cbda6636a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:24.031847Z","signature_b64":"gVlbdRlkMSmyS7xYmLCK038qqiphAIaY7DDpP5OJdE76/RC79ofxtFF7KY74yqD81uZ1qp6qycC2tSl3EeZeAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40602af4276e8d1fe9ed24e8565a43ece57210fc8565150a2c009076c9336ebd","last_reissued_at":"2026-05-18T01:15:24.031123Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:24.031123Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Parking functions, Shi arrangements, and mixed graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Amanda Ruiz, Ana Berrizbeitia, Claudia Rodriguez, Matthias Beck, Michael Dairyko, Schuyler Veeneman","submitted_at":"2014-05-22T01:22:01Z","abstract_excerpt":"The \\emph{Shi arrangement} is the set of all hyperplanes in $\\mathbb R^n$ of the form $x_j - x_k = 0$ or $1$ for $1 \\le j < k \\le n$. Shi observed in 1986 that the number of regions (i.e., connected components of the complement) of this arrangement is $(n+1)^{n-1}$. An unrelated combinatorial concept is that of a \\emph{parking function}, i.e., a sequence $(x_1, x_2, ..., x_n)$ of positive integers that, when rearranged from smallest to largest, satisfies $x_k \\le k$. (There is an illustrative reason for the term \\emph{parking function}.) It turns out that the number of parking functions of len"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.5587","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1405.5587","created_at":"2026-05-18T01:15:24.031237+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.5587v2","created_at":"2026-05-18T01:15:24.031237+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.5587","created_at":"2026-05-18T01:15:24.031237+00:00"},{"alias_kind":"pith_short_12","alias_value":"IBQCV5BHN2GR","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_16","alias_value":"IBQCV5BHN2GR72PN","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_8","alias_value":"IBQCV5BH","created_at":"2026-05-18T12:28:33.132498+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IBQCV5BHN2GR72PNETUFMWSD5T","json":"https://pith.science/pith/IBQCV5BHN2GR72PNETUFMWSD5T.json","graph_json":"https://pith.science/api/pith-number/IBQCV5BHN2GR72PNETUFMWSD5T/graph.json","events_json":"https://pith.science/api/pith-number/IBQCV5BHN2GR72PNETUFMWSD5T/events.json","paper":"https://pith.science/paper/IBQCV5BH"},"agent_actions":{"view_html":"https://pith.science/pith/IBQCV5BHN2GR72PNETUFMWSD5T","download_json":"https://pith.science/pith/IBQCV5BHN2GR72PNETUFMWSD5T.json","view_paper":"https://pith.science/paper/IBQCV5BH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.5587&json=true","fetch_graph":"https://pith.science/api/pith-number/IBQCV5BHN2GR72PNETUFMWSD5T/graph.json","fetch_events":"https://pith.science/api/pith-number/IBQCV5BHN2GR72PNETUFMWSD5T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IBQCV5BHN2GR72PNETUFMWSD5T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IBQCV5BHN2GR72PNETUFMWSD5T/action/storage_attestation","attest_author":"https://pith.science/pith/IBQCV5BHN2GR72PNETUFMWSD5T/action/author_attestation","sign_citation":"https://pith.science/pith/IBQCV5BHN2GR72PNETUFMWSD5T/action/citation_signature","submit_replication":"https://pith.science/pith/IBQCV5BHN2GR72PNETUFMWSD5T/action/replication_record"}},"created_at":"2026-05-18T01:15:24.031237+00:00","updated_at":"2026-05-18T01:15:24.031237+00:00"}