{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:T7VICVXI4JVQTFZB52LLXIY4G6","short_pith_number":"pith:T7VICVXI","schema_version":"1.0","canonical_sha256":"9fea8156e8e26b099721ee96bba31c37a40ae150e617ecb29c53d25e4562dbab","source":{"kind":"arxiv","id":"1504.06383","version":3},"attestation_state":"computed","paper":{"title":"Combinatorics of the zeta map on rational Dyck paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cesar Ceballos, Christopher R. H. Hanusa, Tom Denton","submitted_at":"2015-04-24T03:31:07Z","abstract_excerpt":"An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $\\P$ conjugate is enough to recover $P$.\n  Our method begets an area-preserving involution $\\chi$ on the set of $(a,b)$-Dyck paths when $\\zeta$ is a bijection, as well as a new method for calculating $\\zeta^{-1}$ on classical Dyck paths. For certain nice $(a"},"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":"1504.06383","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-24T03:31:07Z","cross_cats_sorted":[],"title_canon_sha256":"c5b64a1ab804d5434bbbd3da9cf21183d0077d5aa0c759ec9297b327556dab09","abstract_canon_sha256":"7a8e85e7237f5d2815e75a7cc287554756b6cfb48e911b1395d08ab8c0ea6e29"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:26.205597Z","signature_b64":"sycGoVEp95NePtnfWX6LuuG3QuuoY/h22TB2kkEYibFwU5F2ym+t5gQ/iagbRa/NTQ/zUeyla24rg3NC6n4OCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9fea8156e8e26b099721ee96bba31c37a40ae150e617ecb29c53d25e4562dbab","last_reissued_at":"2026-05-18T01:20:26.204979Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:26.204979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Combinatorics of the zeta map on rational Dyck paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cesar Ceballos, Christopher R. H. Hanusa, Tom Denton","submitted_at":"2015-04-24T03:31:07Z","abstract_excerpt":"An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $\\P$ conjugate is enough to recover $P$.\n  Our method begets an area-preserving involution $\\chi$ on the set of $(a,b)$-Dyck paths when $\\zeta$ is a bijection, as well as a new method for calculating $\\zeta^{-1}$ on classical Dyck paths. For certain nice $(a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06383","kind":"arxiv","version":3},"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":"1504.06383","created_at":"2026-05-18T01:20:26.205059+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.06383v3","created_at":"2026-05-18T01:20:26.205059+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.06383","created_at":"2026-05-18T01:20:26.205059+00:00"},{"alias_kind":"pith_short_12","alias_value":"T7VICVXI4JVQ","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"T7VICVXI4JVQTFZB","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"T7VICVXI","created_at":"2026-05-18T12:29:42.218222+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/T7VICVXI4JVQTFZB52LLXIY4G6","json":"https://pith.science/pith/T7VICVXI4JVQTFZB52LLXIY4G6.json","graph_json":"https://pith.science/api/pith-number/T7VICVXI4JVQTFZB52LLXIY4G6/graph.json","events_json":"https://pith.science/api/pith-number/T7VICVXI4JVQTFZB52LLXIY4G6/events.json","paper":"https://pith.science/paper/T7VICVXI"},"agent_actions":{"view_html":"https://pith.science/pith/T7VICVXI4JVQTFZB52LLXIY4G6","download_json":"https://pith.science/pith/T7VICVXI4JVQTFZB52LLXIY4G6.json","view_paper":"https://pith.science/paper/T7VICVXI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.06383&json=true","fetch_graph":"https://pith.science/api/pith-number/T7VICVXI4JVQTFZB52LLXIY4G6/graph.json","fetch_events":"https://pith.science/api/pith-number/T7VICVXI4JVQTFZB52LLXIY4G6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T7VICVXI4JVQTFZB52LLXIY4G6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T7VICVXI4JVQTFZB52LLXIY4G6/action/storage_attestation","attest_author":"https://pith.science/pith/T7VICVXI4JVQTFZB52LLXIY4G6/action/author_attestation","sign_citation":"https://pith.science/pith/T7VICVXI4JVQTFZB52LLXIY4G6/action/citation_signature","submit_replication":"https://pith.science/pith/T7VICVXI4JVQTFZB52LLXIY4G6/action/replication_record"}},"created_at":"2026-05-18T01:20:26.205059+00:00","updated_at":"2026-05-18T01:20:26.205059+00:00"}