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In this paper, using bijections we first derive the counterparts in {\\em Andr\\'e permutations} and {\\em Simsun permutations} for the Entringer numbers $(E_{n,k})$, and then the counterparts in {\\em signed Andr\\'e permutations} and {\\em type $B$ increasing 1-2 trees} for the Arnold numbers $(S_{n,k})$."},"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":"2006.00507","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-05-31T12:39:09Z","cross_cats_sorted":[],"title_canon_sha256":"5a51acc46e8e7f0046fdde0d635a07f5feceee214a7eabdb8298bc92f7abb651","abstract_canon_sha256":"02f9a30131a038c7ec26c909f537379405f6ee9b1677e5001afc6459a8c25605"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:06:40.079839Z","signature_b64":"JE12camqHOYUPDaLOVpYXE4N9DOHAgFSIfb4VY1cMGr7QzCTaP1OmzUU/UuCsKV5wKpwnjm2ztiyv1gaVdWRAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a37060ddef81cea5f3cdb464284509be3b882359f553ee07a060708a41e68ff9","last_reissued_at":"2026-07-05T04:06:40.079424Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:06:40.079424Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"More bijections for Entringer and Arnold families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Heesung Shin, Jiang Zeng","submitted_at":"2020-05-31T12:39:09Z","abstract_excerpt":"The Euler number $E_n$ (resp. 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