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Mao conjectured some additional inequalities. We prove some of Mao's rank inequality conjectures for both the rank and the $M_2$ rank modulo $10$ using elementary methods."},"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":"1509.06039","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-20T18:33:21Z","cross_cats_sorted":[],"title_canon_sha256":"a1d14a9c3b2dd9389203aa55fc2b17db16aa9e94610ae90ddec4688ff6509485","abstract_canon_sha256":"e702eb7faa432e7371fa4ad5141aa031231d98562775d164e34ec9503a0a6e27"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:36.029423Z","signature_b64":"dQcQeFN5+cyvqBlMPAA3toXq923keJ1VHYF7hbqoUWf9uLT3QjGUyWKTtaURGd6uMHQLrazGvdcBNOrT0+pbAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"950bfb302fa5d14c39ec1a3b274ee0f25f7f7dbaa8ea888a4c7e87541a018ec8","last_reissued_at":"2026-05-18T01:32:36.028798Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:36.028798Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A proof of some conjectures of Mao on partition rank inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Elena Iannuzzi, Ethan Alwaise, Holly Swisher","submitted_at":"2015-09-20T18:33:21Z","abstract_excerpt":"Based on work of Atkin and Swinnerton-Dyer on partition rank difference functions, and more recent work of Lovejoy and Osburn, Mao has proved several inequalities between partition ranks modulo $10$, and additional results modulo $6$ and $10$ for the $M_2$ rank of partitions without repeated odd parts. 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