{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1994:KUPPKQJELLYJ56YCTNTWYTKT4V","short_pith_number":"pith:KUPPKQJE","canonical_record":{"source":{"id":"math/9409212","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"1994-09-16T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"39e279346730fd66aa8d22b4f78faeab15805badb3ea36165c21a9c6514dcbba","abstract_canon_sha256":"83f9f54e4113b01fa107543b698cfafd92be4a325ba1da6ddb71e7caf8e6d458"},"schema_version":"1.0"},"canonical_sha256":"551ef541245af09efb029b676c4d53e55792bd6b62618fb90dda6142ab8feeb9","source":{"kind":"arxiv","id":"math/9409212","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9409212","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/9409212v1","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9409212","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"pith_short_12","alias_value":"KUPPKQJELLYJ","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"KUPPKQJELLYJ56YC","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"KUPPKQJE","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1994:KUPPKQJELLYJ56YCTNTWYTKT4V","target":"record","payload":{"canonical_record":{"source":{"id":"math/9409212","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"1994-09-16T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"39e279346730fd66aa8d22b4f78faeab15805badb3ea36165c21a9c6514dcbba","abstract_canon_sha256":"83f9f54e4113b01fa107543b698cfafd92be4a325ba1da6ddb71e7caf8e6d458"},"schema_version":"1.0"},"canonical_sha256":"551ef541245af09efb029b676c4d53e55792bd6b62618fb90dda6142ab8feeb9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:51.119574Z","signature_b64":"Qz/RYwXHRE+RRqTjcxe0tiGKt7bqXL5LdlCExcJjd7PtmwMFsu3wHAnkKwZrR7vQyYOH8kNbzo2gmuUete/HBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"551ef541245af09efb029b676c4d53e55792bd6b62618fb90dda6142ab8feeb9","last_reissued_at":"2026-05-18T01:05:51.118991Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:51.118991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9409212","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nGBWxyHsyFMXeEXV0/vaKj5WCSyEWwa/ve/P4g2e6TutUZyG1Qmmla+LwvPOdrwLvMN2c5vEKjppiBnUCh/BDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T17:36:07.362364Z"},"content_sha256":"8de31e9738c5449d3ba1d87e63764d6a40c9fc6ba4f54d1442b331e1da8d4811","schema_version":"1.0","event_id":"sha256:8de31e9738c5449d3ba1d87e63764d6a40c9fc6ba4f54d1442b331e1da8d4811"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1994:KUPPKQJELLYJ56YCTNTWYTKT4V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting pairs of lattice paths by intersections","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Herbert S. Wilf, Ira Gessel, Lily Yen, Walter Shur, Wayne Goddard","submitted_at":"1994-09-16T00:00:00Z","abstract_excerpt":"On an $r\\times (n-r)$ lattice rectangle, we first consider walks that begin at the SW corner, proceed with unit steps in either of the directions E or N, and terminate at the NE corner of the rectangle. For each integer $k$ we ask for $N_k^{n,r}$, the number of {\\em ordered\\/} pairs of these walks that intersect in exactly $k$ points. The number of points in the intersection of two such walks is defined as the cardinality of the intersection of their two sets of vertices, excluding the initial and terminal vertices. We find two explicit formulas for the numbers $N_k^{n,r}$. Next we note that $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9409212","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1jEylwKGsu6Egm4+czVL8n2OLzSo1Zf8Kx7V0VLR7TFUZCD+HAzwmMTeEN9vn8QP+GL6z+lTcW8OWgB3RfynAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T17:36:07.363059Z"},"content_sha256":"e6f4dfb3a07d6182642896069930aa4f02f84097af16f5a2d01589f28db7a089","schema_version":"1.0","event_id":"sha256:e6f4dfb3a07d6182642896069930aa4f02f84097af16f5a2d01589f28db7a089"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KUPPKQJELLYJ56YCTNTWYTKT4V/bundle.json","state_url":"https://pith.science/pith/KUPPKQJELLYJ56YCTNTWYTKT4V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KUPPKQJELLYJ56YCTNTWYTKT4V/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-09T17:36:07Z","links":{"resolver":"https://pith.science/pith/KUPPKQJELLYJ56YCTNTWYTKT4V","bundle":"https://pith.science/pith/KUPPKQJELLYJ56YCTNTWYTKT4V/bundle.json","state":"https://pith.science/pith/KUPPKQJELLYJ56YCTNTWYTKT4V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KUPPKQJELLYJ56YCTNTWYTKT4V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1994:KUPPKQJELLYJ56YCTNTWYTKT4V","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":"83f9f54e4113b01fa107543b698cfafd92be4a325ba1da6ddb71e7caf8e6d458","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"1994-09-16T00:00:00Z","title_canon_sha256":"39e279346730fd66aa8d22b4f78faeab15805badb3ea36165c21a9c6514dcbba"},"schema_version":"1.0","source":{"id":"math/9409212","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9409212","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/9409212v1","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9409212","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"pith_short_12","alias_value":"KUPPKQJELLYJ","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"KUPPKQJELLYJ56YC","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"KUPPKQJE","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:e6f4dfb3a07d6182642896069930aa4f02f84097af16f5a2d01589f28db7a089","target":"graph","created_at":"2026-05-18T01:05:51Z","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":"On an $r\\times (n-r)$ lattice rectangle, we first consider walks that begin at the SW corner, proceed with unit steps in either of the directions E or N, and terminate at the NE corner of the rectangle. For each integer $k$ we ask for $N_k^{n,r}$, the number of {\\em ordered\\/} pairs of these walks that intersect in exactly $k$ points. The number of points in the intersection of two such walks is defined as the cardinality of the intersection of their two sets of vertices, excluding the initial and terminal vertices. We find two explicit formulas for the numbers $N_k^{n,r}$. Next we note that $","authors_text":"Herbert S. Wilf, Ira Gessel, Lily Yen, Walter Shur, Wayne Goddard","cross_cats":[],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"1994-09-16T00:00:00Z","title":"Counting pairs of lattice paths by intersections"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9409212","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":"sha256:8de31e9738c5449d3ba1d87e63764d6a40c9fc6ba4f54d1442b331e1da8d4811","target":"record","created_at":"2026-05-18T01:05:51Z","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":"83f9f54e4113b01fa107543b698cfafd92be4a325ba1da6ddb71e7caf8e6d458","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"1994-09-16T00:00:00Z","title_canon_sha256":"39e279346730fd66aa8d22b4f78faeab15805badb3ea36165c21a9c6514dcbba"},"schema_version":"1.0","source":{"id":"math/9409212","kind":"arxiv","version":1}},"canonical_sha256":"551ef541245af09efb029b676c4d53e55792bd6b62618fb90dda6142ab8feeb9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"551ef541245af09efb029b676c4d53e55792bd6b62618fb90dda6142ab8feeb9","first_computed_at":"2026-05-18T01:05:51.118991Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:51.118991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Qz/RYwXHRE+RRqTjcxe0tiGKt7bqXL5LdlCExcJjd7PtmwMFsu3wHAnkKwZrR7vQyYOH8kNbzo2gmuUete/HBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:51.119574Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9409212","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8de31e9738c5449d3ba1d87e63764d6a40c9fc6ba4f54d1442b331e1da8d4811","sha256:e6f4dfb3a07d6182642896069930aa4f02f84097af16f5a2d01589f28db7a089"],"state_sha256":"23abd6fa2d9d4bf67d0da376dcc04a06a495016b60c9c239bec42f6d4d1d1e0f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b+oxroX8/eJX1zfO/TcLhFEj5g4mayzcdyPAq9N7jljyKc0UBzi9SknbGI3dBpa+wqdqKvGDHztdPiUfbY69CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T17:36:07.367283Z","bundle_sha256":"122781ddac5af5d1997e0dd31f669a2348010eaffb62d6293db495a768d85747"}}