{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:FGK4P43H57WRUY4G4WK6M4DKIM","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":"699b2fed02d01f7b55a7152084dd934825f6a8d27d14c390aecc1bd5867991ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-31T18:29:15Z","title_canon_sha256":"3e2d6aa3af4231db5fc72eec5afd556959c457bf0c7c4e6d2359bfdf3c5f0654"},"schema_version":"1.0","source":{"id":"1301.7714","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.7714","created_at":"2026-05-18T03:34:48Z"},{"alias_kind":"arxiv_version","alias_value":"1301.7714v1","created_at":"2026-05-18T03:34:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.7714","created_at":"2026-05-18T03:34:48Z"},{"alias_kind":"pith_short_12","alias_value":"FGK4P43H57WR","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"FGK4P43H57WRUY4G","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"FGK4P43H","created_at":"2026-05-18T12:27:45Z"}],"graph_snapshots":[{"event_id":"sha256:6fa469c9b9242bb3913e43bb0071e1e08268409a76bf3c0e79c514dd58daef3b","target":"graph","created_at":"2026-05-18T03:34:48Z","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":"Let M(n,k,r,s) be the number of ordered paths in the plane, with unit steps E or N, that intersect k times in which the first path ends at the point (r,n-r) and the second path ends at the point (s,n-s). Our main object of study in this paper is the sum of the numbers M(n,k,r,s) over r and s where r+s is fixed. We consider even and odd values of r+s separately, and we derive a simpler formula for M(n,k,r,s) than previously appeared in the literature.","authors_text":"Ira M. Gessel, Walter Shur","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-31T18:29:15Z","title":"Even and Odd Pairs of Lattice Paths with Multiple Intersections"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.7714","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":"sha256:9ed0c2166e412e876f062c951862ff97f3bbfab0064ef0f4e8f1043210c0311f","target":"record","created_at":"2026-05-18T03:34:48Z","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":"699b2fed02d01f7b55a7152084dd934825f6a8d27d14c390aecc1bd5867991ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-31T18:29:15Z","title_canon_sha256":"3e2d6aa3af4231db5fc72eec5afd556959c457bf0c7c4e6d2359bfdf3c5f0654"},"schema_version":"1.0","source":{"id":"1301.7714","kind":"arxiv","version":1}},"canonical_sha256":"2995c7f367efed1a6386e595e6706a4312194f846d73d27e017db2ecd6870922","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2995c7f367efed1a6386e595e6706a4312194f846d73d27e017db2ecd6870922","first_computed_at":"2026-05-18T03:34:48.861976Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:48.861976Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wrRGMmPsTwJtSNuIlNB6TyNyErPPJ6Mfa+8irxQ+60zF1PlghmSNKqQ3taZgcy678Z49KTVuJ2ZmN4V6NCimBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:48.862544Z","signed_message":"canonical_sha256_bytes"},"source_id":"1301.7714","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9ed0c2166e412e876f062c951862ff97f3bbfab0064ef0f4e8f1043210c0311f","sha256:6fa469c9b9242bb3913e43bb0071e1e08268409a76bf3c0e79c514dd58daef3b"],"state_sha256":"53655d4afa74b8c5fd8758f45230ee65f102e9366152683713d15754ce69e58d"}