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Chan and Mao proved that for any nonnegative integers $m$ and $n$, $N(m,n)\\geq N(m+2,n)$ and for any nonnegative integers $m$ and $n$ such that $n\\geq12$, $n\\neq m+2$, $N(m,n)\\geq N(m,n-1)$. Recently, Ji and Zang showed that for $n\\geq 44$ and $1\\leq m\\leq n-1$, $M(m-1,n)\\geq M(m,n)$ and for $n\\geq 14$ and $0\\leq m\\leq n-2$, $M(m,n)\\geq M(m,n-1)$. In this paper, we analogue the result of Ji and Zang to overpartitions. Note that Bringmann, Lovejoy and Osburn"},"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":"1811.10013","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-25T13:51:31Z","cross_cats_sorted":[],"title_canon_sha256":"f9c190a7fcb861256dd5cb9bd57588e53bd71e5f4952f12ce1a8d351838f7275","abstract_canon_sha256":"d0ded73b899dd3b2db7e5cfc5cc2e90adb7f51ac126b13cf5fd25a6e870c9d23"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:58.872042Z","signature_b64":"kZtSIJFXtmTReFH+XdQjGgO5eyb2/khkPaqNMIxwmskSRj52HUIulovPzqJD758NfJVqjo29f2lJVZcrRGPOCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"013a219e66d5146d40708972dd00f0a06bd5a9f1d64eb4ca13b4daac171c4f91","last_reissued_at":"2026-05-17T23:59:58.871563Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:58.871563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Unimodality of the Crank on Overpartitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Helen W.J. Zhang, Wenston J.T. Zang","submitted_at":"2018-11-25T13:51:31Z","abstract_excerpt":"Let $N(m,n)$ denote the number of partitions of $n$ with rank $m$, and let $M(m,n)$ denote the number of partitions of $n$ with crank $m$. Chan and Mao proved that for any nonnegative integers $m$ and $n$, $N(m,n)\\geq N(m+2,n)$ and for any nonnegative integers $m$ and $n$ such that $n\\geq12$, $n\\neq m+2$, $N(m,n)\\geq N(m,n-1)$. Recently, Ji and Zang showed that for $n\\geq 44$ and $1\\leq m\\leq n-1$, $M(m-1,n)\\geq M(m,n)$ and for $n\\geq 14$ and $0\\leq m\\leq n-2$, $M(m,n)\\geq M(m,n-1)$. In this paper, we analogue the result of Ji and Zang to overpartitions. 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