{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:OUCUGFHRDXDACJGOE73NRURIJW","short_pith_number":"pith:OUCUGFHR","schema_version":"1.0","canonical_sha256":"75054314f11dc60124ce27f6d8d2284db1aff3aef99d6cc39a089b6b530038e1","source":{"kind":"arxiv","id":"1908.03912","version":1},"attestation_state":"computed","paper":{"title":"Bijective recurrences concerning two Schr\\\"oder triangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Shishuo Fu, Yaling Wang","submitted_at":"2019-08-11T14:23:44Z","abstract_excerpt":"Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schr\\\"oder paths (resp. little Schr\\\"oder paths) of length $2n$ with $k$ hills, and set $r(0,0)=s(0,0)=1$. We bijectively establish the following recurrence relations: \\begin{align*} r(n,0)&=\\sum\\limits_{j=0}^{n-1}2^{j}r(n-1,j), r(n,k)&=r(n-1,k-1)+\\sum\\limits_{j=k}^{n-1}2^{j-k}r(n-1,j),\\quad 1\\le k\\le n, s(n,0) &=\\sum\\limits_{j=1}^{n-1}2\\cdot3^{j-1}s(n-1,j), s(n,k) &=s(n-1,k-1)+\\sum\\limits_{j=k+1}^{n-1}2\\cdot3^{j-k-1}s(n-1,j),\\quad 1\\le k\\le n. \\end{align*} The infinite lower triangular matrices $[r(n,k)]_{n,k\\ge 0}$ and $[s(n,k)]_{n,k\\ge 0}$, who"},"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":"1908.03912","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-11T14:23:44Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"0c63a072cca47ba0d81b4c6a4569525b65d79f05dd9a423329f62d13130a718a","abstract_canon_sha256":"4384004a62c17c9ef7e1c8689ca9d36d6116ebc4422903f4dfcfed90addfb61a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:53:19.329214Z","signature_b64":"KpPSGzQleD0d/h5JDDgSGk4RunWaii3X5m1fT7L5yEhGLWCpNTJbWdPXnBT4RBGvRda5NzqLb7iq3V+hsTjpAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"75054314f11dc60124ce27f6d8d2284db1aff3aef99d6cc39a089b6b530038e1","last_reissued_at":"2026-07-04T23:53:19.328852Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:53:19.328852Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bijective recurrences concerning two Schr\\\"oder triangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Shishuo Fu, Yaling Wang","submitted_at":"2019-08-11T14:23:44Z","abstract_excerpt":"Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schr\\\"oder paths (resp. little Schr\\\"oder paths) of length $2n$ with $k$ hills, and set $r(0,0)=s(0,0)=1$. We bijectively establish the following recurrence relations: \\begin{align*} r(n,0)&=\\sum\\limits_{j=0}^{n-1}2^{j}r(n-1,j), r(n,k)&=r(n-1,k-1)+\\sum\\limits_{j=k}^{n-1}2^{j-k}r(n-1,j),\\quad 1\\le k\\le n, s(n,0) &=\\sum\\limits_{j=1}^{n-1}2\\cdot3^{j-1}s(n-1,j), s(n,k) &=s(n-1,k-1)+\\sum\\limits_{j=k+1}^{n-1}2\\cdot3^{j-k-1}s(n-1,j),\\quad 1\\le k\\le n. \\end{align*} The infinite lower triangular matrices $[r(n,k)]_{n,k\\ge 0}$ and $[s(n,k)]_{n,k\\ge 0}$, who"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03912","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.03912/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1908.03912","created_at":"2026-07-04T23:53:19.328905+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.03912v1","created_at":"2026-07-04T23:53:19.328905+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.03912","created_at":"2026-07-04T23:53:19.328905+00:00"},{"alias_kind":"pith_short_12","alias_value":"OUCUGFHRDXDA","created_at":"2026-07-04T23:53:19.328905+00:00"},{"alias_kind":"pith_short_16","alias_value":"OUCUGFHRDXDACJGO","created_at":"2026-07-04T23:53:19.328905+00:00"},{"alias_kind":"pith_short_8","alias_value":"OUCUGFHR","created_at":"2026-07-04T23:53:19.328905+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/OUCUGFHRDXDACJGOE73NRURIJW","json":"https://pith.science/pith/OUCUGFHRDXDACJGOE73NRURIJW.json","graph_json":"https://pith.science/api/pith-number/OUCUGFHRDXDACJGOE73NRURIJW/graph.json","events_json":"https://pith.science/api/pith-number/OUCUGFHRDXDACJGOE73NRURIJW/events.json","paper":"https://pith.science/paper/OUCUGFHR"},"agent_actions":{"view_html":"https://pith.science/pith/OUCUGFHRDXDACJGOE73NRURIJW","download_json":"https://pith.science/pith/OUCUGFHRDXDACJGOE73NRURIJW.json","view_paper":"https://pith.science/paper/OUCUGFHR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.03912&json=true","fetch_graph":"https://pith.science/api/pith-number/OUCUGFHRDXDACJGOE73NRURIJW/graph.json","fetch_events":"https://pith.science/api/pith-number/OUCUGFHRDXDACJGOE73NRURIJW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OUCUGFHRDXDACJGOE73NRURIJW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OUCUGFHRDXDACJGOE73NRURIJW/action/storage_attestation","attest_author":"https://pith.science/pith/OUCUGFHRDXDACJGOE73NRURIJW/action/author_attestation","sign_citation":"https://pith.science/pith/OUCUGFHRDXDACJGOE73NRURIJW/action/citation_signature","submit_replication":"https://pith.science/pith/OUCUGFHRDXDACJGOE73NRURIJW/action/replication_record"}},"created_at":"2026-07-04T23:53:19.328905+00:00","updated_at":"2026-07-04T23:53:19.328905+00:00"}