{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:OUCUGFHRDXDACJGOE73NRURIJW","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":"4384004a62c17c9ef7e1c8689ca9d36d6116ebc4422903f4dfcfed90addfb61a","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-11T14:23:44Z","title_canon_sha256":"0c63a072cca47ba0d81b4c6a4569525b65d79f05dd9a423329f62d13130a718a"},"schema_version":"1.0","source":{"id":"1908.03912","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.03912","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"arxiv_version","alias_value":"1908.03912v1","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.03912","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"pith_short_12","alias_value":"OUCUGFHRDXDA","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"pith_short_16","alias_value":"OUCUGFHRDXDACJGO","created_at":"2026-07-04T23:53:19Z"},{"alias_kind":"pith_short_8","alias_value":"OUCUGFHR","created_at":"2026-07-04T23:53:19Z"}],"graph_snapshots":[{"event_id":"sha256:1c96c93b8659aa24aa25a1cc440c7e0557b8ee6055ff6d12127dab81b35f1784","target":"graph","created_at":"2026-07-04T23:53:19Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.03912/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Shishuo Fu, Yaling Wang","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-11T14:23:44Z","title":"Bijective recurrences concerning two Schr\\\"oder triangles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03912","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:d6406272b2447ee07600661ffca6881aef8fd41ecbb5da7e52f66f3d00a765de","target":"record","created_at":"2026-07-04T23:53:19Z","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":"4384004a62c17c9ef7e1c8689ca9d36d6116ebc4422903f4dfcfed90addfb61a","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-11T14:23:44Z","title_canon_sha256":"0c63a072cca47ba0d81b4c6a4569525b65d79f05dd9a423329f62d13130a718a"},"schema_version":"1.0","source":{"id":"1908.03912","kind":"arxiv","version":1}},"canonical_sha256":"75054314f11dc60124ce27f6d8d2284db1aff3aef99d6cc39a089b6b530038e1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"75054314f11dc60124ce27f6d8d2284db1aff3aef99d6cc39a089b6b530038e1","first_computed_at":"2026-07-04T23:53:19.328852Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:53:19.328852Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KpPSGzQleD0d/h5JDDgSGk4RunWaii3X5m1fT7L5yEhGLWCpNTJbWdPXnBT4RBGvRda5NzqLb7iq3V+hsTjpAg==","signature_status":"signed_v1","signed_at":"2026-07-04T23:53:19.329214Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.03912","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d6406272b2447ee07600661ffca6881aef8fd41ecbb5da7e52f66f3d00a765de","sha256:1c96c93b8659aa24aa25a1cc440c7e0557b8ee6055ff6d12127dab81b35f1784"],"state_sha256":"81b71b1cc1501620eab6358c068b19448a3c3672dd71211c26edaeabbc7f0541"}