{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:XDV3PO5V3VZU2SZRIUTV3VA64G","short_pith_number":"pith:XDV3PO5V","schema_version":"1.0","canonical_sha256":"b8ebb7bbb5dd734d4b3145275dd41ee1a27489b528acfdfe7d39b870cd9ac51a","source":{"kind":"arxiv","id":"1601.07988","version":3},"attestation_state":"computed","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 $A \\subseteq [n]$ "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1601.07988","created_at":"2026-05-18T00:36:48.032646+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.07988v3","created_at":"2026-05-18T00:36:48.032646+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.07988","created_at":"2026-05-18T00:36:48.032646+00:00"},{"alias_kind":"pith_short_12","alias_value":"XDV3PO5V3VZU","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_16","alias_value":"XDV3PO5V3VZU2SZR","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_8","alias_value":"XDV3PO5V","created_at":"2026-05-18T12:30:51.357362+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G","json":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G.json","graph_json":"https://pith.science/api/pith-number/XDV3PO5V3VZU2SZRIUTV3VA64G/graph.json","events_json":"https://pith.science/api/pith-number/XDV3PO5V3VZU2SZRIUTV3VA64G/events.json","paper":"https://pith.science/paper/XDV3PO5V"},"agent_actions":{"view_html":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G","download_json":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G.json","view_paper":"https://pith.science/paper/XDV3PO5V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.07988&json=true","fetch_graph":"https://pith.science/api/pith-number/XDV3PO5V3VZU2SZRIUTV3VA64G/graph.json","fetch_events":"https://pith.science/api/pith-number/XDV3PO5V3VZU2SZRIUTV3VA64G/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G/action/storage_attestation","attest_author":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G/action/author_attestation","sign_citation":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G/action/citation_signature","submit_replication":"https://pith.science/pith/XDV3PO5V3VZU2SZRIUTV3VA64G/action/replication_record"}},"created_at":"2026-05-18T00:36:48.032646+00:00","updated_at":"2026-05-18T00:36:48.032646+00:00"}