{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:OHZEFSVUOVJ3MREDYCL5V4E5CI","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":"f009d128dc198e1933899564aa37aa9894ec4cb68e0b0c37afb23087fd81cae4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-25T16:10:15Z","title_canon_sha256":"83a4ab7502f34a8ca83dca549a0086c9da90f945b54e65d30adff61814dc7c43"},"schema_version":"1.0","source":{"id":"1306.6015","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.6015","created_at":"2026-05-18T03:19:57Z"},{"alias_kind":"arxiv_version","alias_value":"1306.6015v1","created_at":"2026-05-18T03:19:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.6015","created_at":"2026-05-18T03:19:57Z"},{"alias_kind":"pith_short_12","alias_value":"OHZEFSVUOVJ3","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"OHZEFSVUOVJ3MRED","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"OHZEFSVU","created_at":"2026-05-18T12:27:54Z"}],"graph_snapshots":[{"event_id":"sha256:faa0fc8048e1ae2fc49c9f25f7203d3e3a658ac4c50209db6c71928b93a7cce1","target":"graph","created_at":"2026-05-18T03:19:57Z","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":"Koroljuk gave a summation formula for counting the number of lattice paths from $(0,0)$ to $(m,n)$ with $(1,0), (0,1)$-steps in the plane that stay strictly above the line $y=k(x-d)$, where $k$ and $d$ are positive integers. In this paper we obtain an explicit formula for the number of lattice paths from $(a,b)$ to $(m,n)$ above the diagonal $y=kx-r$, where $r$ is a rational number. Our result slightly generalizes Koroljuk's formula, while the former can be essentially derived from the latter. However, our proof uses a recurrence with respect to the starting points, and hereby presents a new a","authors_text":"James J.Y. Zhao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-25T16:10:15Z","title":"Koroljuk's formula for counting lattice paths revisited"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.6015","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:a3de9e9cb38addece344fb7298c4013fb17916c5ed73a021c2a095267b853518","target":"record","created_at":"2026-05-18T03:19:57Z","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":"f009d128dc198e1933899564aa37aa9894ec4cb68e0b0c37afb23087fd81cae4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-25T16:10:15Z","title_canon_sha256":"83a4ab7502f34a8ca83dca549a0086c9da90f945b54e65d30adff61814dc7c43"},"schema_version":"1.0","source":{"id":"1306.6015","kind":"arxiv","version":1}},"canonical_sha256":"71f242cab47553b64483c097daf09d1201d210cd65c10e19c7dd3b8524d6db18","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"71f242cab47553b64483c097daf09d1201d210cd65c10e19c7dd3b8524d6db18","first_computed_at":"2026-05-18T03:19:57.977515Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:19:57.977515Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wddMlikE8dkLueHQpmA/NeG3x5nTx0vNQdlSp0DUKcW82yuRIz0G1ZQhOg58DVclCK1pHW/04jfM1HGFDJ4eBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:19:57.977984Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.6015","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a3de9e9cb38addece344fb7298c4013fb17916c5ed73a021c2a095267b853518","sha256:faa0fc8048e1ae2fc49c9f25f7203d3e3a658ac4c50209db6c71928b93a7cce1"],"state_sha256":"98a277bb836e0748fc25a3e752e68d6f2f6cece64b5762616aa5e292a9e7a9fa"}