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Equivalently, an up-up-or-down-down permutation is one in which every peak and every valley is odd."},"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":"2411.16113","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-25T05:58:21Z","cross_cats_sorted":[],"title_canon_sha256":"5ba3bdcaa670e0095102eb85cec0437387eb0b2aeba5a5fcfbe7ce1473ffd305","abstract_canon_sha256":"371ae35f751ff239189ac794460255ca17227c24cd945349e01e42bdf178365c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:40:04.340604Z","signature_b64":"0MdwY/ZTnLryQlNNs/xFpZjNTSF6mgcnlWTai1I8BLtVhpZYgS3wxOC00lgayr01QqVin/VvOfGTt6bSWJ3iCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c9eda396c81b74d33db9fbcb199fcc1fb90a9c1d84c3406bcffb4629402eb24","last_reissued_at":"2026-07-05T09:40:04.340156Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:40:04.340156Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting up-up-or-down-down permutations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ira M. 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