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More specifically, suppose $A$ is an $n \\times n$ matrix over $\\mathbb{C}$ (resp. $\\mathbb{R}$), and let $\\mathcal{P}$ denote the set of $n \\times n$ matrices over $\\mathbb{C}$ (resp. $\\mathbb{R}$) that can be written as a permutation matrix times a unitary diagonal matrix. Then it is known that the permanent of $A$ satisfies $|\\text{perm}(A)| \\leq \\Vert A \\Vert_{2} ^n$ with equality iff $A/ \\Vert A \\Vert_{2} \\in \\mathcal{P}$ (where $\\Vert A \\Vert_2$ is the operator $2$-norm of"},"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":"1606.07474","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-23T20:54:47Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"7fbe58f29fc6df31f7e988c408eba83cf1ca66d5466b20f19129194e3061e8ef","abstract_canon_sha256":"1a8380c2c5eb27aa69424d53de4d2ef8bdb201a18c3db325c0195133a10fdb33"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:58.050980Z","signature_b64":"J/0gW7lyPE7hEFAYGL6e5w7WCVKMPv+nwTQIbDvs61xdft2AGV80DlH72raTp9J8iEw7Ng6JP3s8hdEE0VYaCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2effb900fdfe5fde675a45716602594d03cab1b901843807c3adff5ff15f1874","last_reissued_at":"2026-05-18T01:11:58.050646Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:58.050646Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A stability result using the matrix norm to bound the permanent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Pat Devlin, Ross Berkowitz","submitted_at":"2016-06-23T20:54:47Z","abstract_excerpt":"We prove a stability version of a general result that bounds the permanent of a matrix in terms of its operator norm. 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