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In this paper we show that both the partition function $\\{p(n+N_0)\\}_{n\\geq 0}$ and the overpartition function $\\{\\overline{p}(n+\\overline{N}_0)\\}_{n\\ge 0}$ satisfy the Briggs inequality for some $N_0$ and $\\overline{N}_{0}$. Based on Chern's formula for $\\eta$-quotients, we further prove that the $k$-regular partition function $\\{p_k(n+N_{k})\\}_{n\\geq 0}$ and the $k$-regular overpartition function $\\{\\overline{p}_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":"2408.16185","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2024-08-29T00:49:36Z","cross_cats_sorted":[],"title_canon_sha256":"7ae1e6eb169308e188ac47e8e5b62f71aa3d75138733b443b64e39581b2a1891","abstract_canon_sha256":"8dea6218dd659228f9dfcdfa89000411c3eadc1dbe5688ed56c1a39549220849"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:00:27.307642Z","signature_b64":"rvnOodBV6MFEHruHRXcz5wgNeDXQxzqMc6QgrhlYYy7dULGMAy92rfYLjtHL7Z8RPd4PTPT8hacDLtttVzAqCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98c65ba0eed03deed45c8c382e8d116bbe3f147410b2619bb961a2ff9e85756c","last_reissued_at":"2026-07-05T09:00:27.307179Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:00:27.307179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Briggs inequality for partitions and overpartitions","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xin-Bei Liu, Zhong-Xue Zhang","submitted_at":"2024-08-29T00:49:36Z","abstract_excerpt":"A sequence of $\\{a_n\\}_{n\\ge 0}$ satisfies the Briggs inequality if \\begin{align*} a_n^2(a_n^2-a_{n-1}a_{n+1})>a_{n-1}^2(a_{n+1}^2-a_na_{n+2}) \\end{align*} holds for any $n\\ge 1$. In this paper we show that both the partition function $\\{p(n+N_0)\\}_{n\\geq 0}$ and the overpartition function $\\{\\overline{p}(n+\\overline{N}_0)\\}_{n\\ge 0}$ satisfy the Briggs inequality for some $N_0$ and $\\overline{N}_{0}$. Based on Chern's formula for $\\eta$-quotients, we further prove that the $k$-regular partition function $\\{p_k(n+N_{k})\\}_{n\\geq 0}$ and the $k$-regular overpartition function $\\{\\overline{p}_k("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.16185","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.16185/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2408.16185","created_at":"2026-07-05T09:00:27.307241+00:00"},{"alias_kind":"arxiv_version","alias_value":"2408.16185v1","created_at":"2026-07-05T09:00:27.307241+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.16185","created_at":"2026-07-05T09:00:27.307241+00:00"},{"alias_kind":"pith_short_12","alias_value":"TDDFXIHO2A66","created_at":"2026-07-05T09:00:27.307241+00:00"},{"alias_kind":"pith_short_16","alias_value":"TDDFXIHO2A665VC4","created_at":"2026-07-05T09:00:27.307241+00:00"},{"alias_kind":"pith_short_8","alias_value":"TDDFXIHO","created_at":"2026-07-05T09:00:27.307241+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TDDFXIHO2A665VC4RQ4C5DIRNO","json":"https://pith.science/pith/TDDFXIHO2A665VC4RQ4C5DIRNO.json","graph_json":"https://pith.science/api/pith-number/TDDFXIHO2A665VC4RQ4C5DIRNO/graph.json","events_json":"https://pith.science/api/pith-number/TDDFXIHO2A665VC4RQ4C5DIRNO/events.json","paper":"https://pith.science/paper/TDDFXIHO"},"agent_actions":{"view_html":"https://pith.science/pith/TDDFXIHO2A665VC4RQ4C5DIRNO","download_json":"https://pith.science/pith/TDDFXIHO2A665VC4RQ4C5DIRNO.json","view_paper":"https://pith.science/paper/TDDFXIHO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2408.16185&json=true","fetch_graph":"https://pith.science/api/pith-number/TDDFXIHO2A665VC4RQ4C5DIRNO/graph.json","fetch_events":"https://pith.science/api/pith-number/TDDFXIHO2A665VC4RQ4C5DIRNO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TDDFXIHO2A665VC4RQ4C5DIRNO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TDDFXIHO2A665VC4RQ4C5DIRNO/action/storage_attestation","attest_author":"https://pith.science/pith/TDDFXIHO2A665VC4RQ4C5DIRNO/action/author_attestation","sign_citation":"https://pith.science/pith/TDDFXIHO2A665VC4RQ4C5DIRNO/action/citation_signature","submit_replication":"https://pith.science/pith/TDDFXIHO2A665VC4RQ4C5DIRNO/action/replication_record"}},"created_at":"2026-07-05T09:00:27.307241+00:00","updated_at":"2026-07-05T09:00:27.307241+00:00"}