{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:XDV3PO5V3VZU2SZRIUTV3VA64G","short_pith_number":"pith:XDV3PO5V","canonical_record":{"source":{"id":"1601.07988","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-29T07:10:28Z","cross_cats_sorted":[],"title_canon_sha256":"d867d9b5394a70172168d5b3d144b6bd11369ef17332c738265eef21840a1dc8","abstract_canon_sha256":"6bc288c4e1712f5bc22e43f3734cd032d24b19f3a8ea569b549e1eb674e037cc"},"schema_version":"1.0"},"canonical_sha256":"b8ebb7bbb5dd734d4b3145275dd41ee1a27489b528acfdfe7d39b870cd9ac51a","source":{"kind":"arxiv","id":"1601.07988","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.07988","created_at":"2026-05-18T00:36:48Z"},{"alias_kind":"arxiv_version","alias_value":"1601.07988v3","created_at":"2026-05-18T00:36:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.07988","created_at":"2026-05-18T00:36:48Z"},{"alias_kind":"pith_short_12","alias_value":"XDV3PO5V3VZU","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XDV3PO5V3VZU2SZR","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XDV3PO5V","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:XDV3PO5V3VZU2SZRIUTV3VA64G","target":"record","payload":{"canonical_record":{"source":{"id":"1601.07988","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-29T07:10:28Z","cross_cats_sorted":[],"title_canon_sha256":"d867d9b5394a70172168d5b3d144b6bd11369ef17332c738265eef21840a1dc8","abstract_canon_sha256":"6bc288c4e1712f5bc22e43f3734cd032d24b19f3a8ea569b549e1eb674e037cc"},"schema_version":"1.0"},"canonical_sha256":"b8ebb7bbb5dd734d4b3145275dd41ee1a27489b528acfdfe7d39b870cd9ac51a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:48.033031Z","signature_b64":"oVyGXyPa/DHQ4U+sawGDBMHpq/WQTRaoqM1xZiWD/et/zlLT7TiZuYwWx1oVF73dOwtU29iLv6T3HyosLCx1Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8ebb7bbb5dd734d4b3145275dd41ee1a27489b528acfdfe7d39b870cd9ac51a","last_reissued_at":"2026-05-18T00:36:48.032569Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:48.032569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.07988","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:36:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d+bdRIJ9vaeBQIe1CPdj6tWxsB6BLL4RetAVEfGI1WQA0MhMGsVU2tAwBASOwWgvIaFbGC8D+QceCxcyyRfXDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T16:55:43.883588Z"},"content_sha256":"3fa199f79a55263cf04fa942f96c0ba0f94199065f81c7da10f7bde381c891a7","schema_version":"1.0","event_id":"sha256:3fa199f79a55263cf04fa942f96c0ba0f94199065f81c7da10f7bde381c891a7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:XDV3PO5V3VZU2SZRIUTV3VA64G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Paired patterns in lattice paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jeffrey B. Remmel, Ran Pan","submitted_at":"2016-01-29T07:10:28Z","abstract_excerpt":"Let $\\mathcal{L}_n$ denote the set of all paths from $[0,0]$ to $[n, n]$ which consist of either unit north steps $N$ or unit east steps $E$ or, equivalently, the set of all words $L \\in \\{E,N\\}^*$ with $n$ $E$'s and $n$ $N$'s. Given $L \\in \\mathcal{L}_n$ and a subset $A$ of $[n] = \\{1, \\ldots, n\\}$, we let $ps_{L}(A)$ denote the word that results from $L$ by removing the $i^{th}$ occurrence of $E$ and the $i^{th}$ occurrence of $N$ in $L$ for all $i \\in [n]-A$, reading from left to right. Then we say that a paired pattern $P \\in \\mathcal{L}_k$ occurs in $L$ if there is some $A \\subseteq [n]$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07988","kind":"arxiv","version":3},"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":"2026-05-18T00:36:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JBYuFR9RYN2slmZ9dx6H1fDOyGhJS7PWyytMxOjSLfQiuqIR/Vr0mxQe9W6c7woRn+87yU2wLGRs+EKJynCvCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T16:55:43.884357Z"},"content_sha256":"79dbf7be5f1acbf7e1fbb651c37c70d22161e7ba2de37f8750dbfc673abb771c","schema_version":"1.0","event_id":"sha256:79dbf7be5f1acbf7e1fbb651c37c70d22161e7ba2de37f8750dbfc673abb771c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G/bundle.json","state_url":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XDV3PO5V3VZU2SZRIUTV3VA64G/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":"2026-06-10T16:55:43Z","links":{"resolver":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G","bundle":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G/bundle.json","state":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XDV3PO5V3VZU2SZRIUTV3VA64G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XDV3PO5V3VZU2SZRIUTV3VA64G","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":"6bc288c4e1712f5bc22e43f3734cd032d24b19f3a8ea569b549e1eb674e037cc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-29T07:10:28Z","title_canon_sha256":"d867d9b5394a70172168d5b3d144b6bd11369ef17332c738265eef21840a1dc8"},"schema_version":"1.0","source":{"id":"1601.07988","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.07988","created_at":"2026-05-18T00:36:48Z"},{"alias_kind":"arxiv_version","alias_value":"1601.07988v3","created_at":"2026-05-18T00:36:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.07988","created_at":"2026-05-18T00:36:48Z"},{"alias_kind":"pith_short_12","alias_value":"XDV3PO5V3VZU","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XDV3PO5V3VZU2SZR","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XDV3PO5V","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:79dbf7be5f1acbf7e1fbb651c37c70d22161e7ba2de37f8750dbfc673abb771c","target":"graph","created_at":"2026-05-18T00:36:48Z","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 $\\mathcal{L}_n$ denote the set of all paths from $[0,0]$ to $[n, n]$ which consist of either unit north steps $N$ or unit east steps $E$ or, equivalently, the set of all words $L \\in \\{E,N\\}^*$ with $n$ $E$'s and $n$ $N$'s. Given $L \\in \\mathcal{L}_n$ and a subset $A$ of $[n] = \\{1, \\ldots, n\\}$, we let $ps_{L}(A)$ denote the word that results from $L$ by removing the $i^{th}$ occurrence of $E$ and the $i^{th}$ occurrence of $N$ in $L$ for all $i \\in [n]-A$, reading from left to right. Then we say that a paired pattern $P \\in \\mathcal{L}_k$ occurs in $L$ if there is some $A \\subseteq [n]$ ","authors_text":"Jeffrey B. Remmel, Ran Pan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-29T07:10:28Z","title":"Paired patterns in lattice paths"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07988","kind":"arxiv","version":3},"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":"sha256:3fa199f79a55263cf04fa942f96c0ba0f94199065f81c7da10f7bde381c891a7","target":"record","created_at":"2026-05-18T00:36:48Z","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":"6bc288c4e1712f5bc22e43f3734cd032d24b19f3a8ea569b549e1eb674e037cc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-29T07:10:28Z","title_canon_sha256":"d867d9b5394a70172168d5b3d144b6bd11369ef17332c738265eef21840a1dc8"},"schema_version":"1.0","source":{"id":"1601.07988","kind":"arxiv","version":3}},"canonical_sha256":"b8ebb7bbb5dd734d4b3145275dd41ee1a27489b528acfdfe7d39b870cd9ac51a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b8ebb7bbb5dd734d4b3145275dd41ee1a27489b528acfdfe7d39b870cd9ac51a","first_computed_at":"2026-05-18T00:36:48.032569Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:36:48.032569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oVyGXyPa/DHQ4U+sawGDBMHpq/WQTRaoqM1xZiWD/et/zlLT7TiZuYwWx1oVF73dOwtU29iLv6T3HyosLCx1Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:36:48.033031Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.07988","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3fa199f79a55263cf04fa942f96c0ba0f94199065f81c7da10f7bde381c891a7","sha256:79dbf7be5f1acbf7e1fbb651c37c70d22161e7ba2de37f8750dbfc673abb771c"],"state_sha256":"ecb1652890d5f54eb8daea03ade865ff79e3e0ceb36d9e90251020fb51a80cd8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LcKvUpkbnuonU9YJRY47C4M/j5GBd8JB5yi/dp1Qw+GR28ykHB2P1S2tYZhJcM39DzAMZ8nV/aEMx7Hafx/9Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T16:55:43.888720Z","bundle_sha256":"67f8ea9af82765dce22b12cacf0a58c592fe8f6490d5e3b506aef3759679e31b"}}