{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:HQQYNAEGHZQ3U5IR43AUBIOJJW","short_pith_number":"pith:HQQYNAEG","schema_version":"1.0","canonical_sha256":"3c218680863e61ba7511e6c140a1c94d9faae88179e7ffd09328322a17d1f17b","source":{"kind":"arxiv","id":"1805.03780","version":1},"attestation_state":"computed","paper":{"title":"$M_2$-Ranks of overpartitions modulo $6$ and $10$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Helen W.J. Zhang","submitted_at":"2018-05-10T01:49:47Z","abstract_excerpt":"In this paper, we obtain inequalities on $M_2$-ranks of overpartitions modulo $6$. Let $\\overline{N}_2(s,m,n)$ to be the number of overpartitions of $n$ whose $M_2$-rank is congruent to $s$ modulo $m$. For $M_2$-ranks modulo $3$, Lovejoy and Osburn derived the generating function of $\\overline{N}_2(s,3,n)-\\overline{N}_2(t,3,n)$, which implies the inequalities $\\overline{N}_2(0,3,n)\\geq\\overline{N}_2(1,3,n)$. For $\\ell=6, 10$, we consider the generating function $\\overline{R}_{s,t}(d,\\ell)$ of the $M_2$-rank differences $\\overline{N}_2(s,\\ell,\\ell n/2+d) + \\overline{N}_2(s+1,\\ell,\\ell n/2+d) - "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1805.03780","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-10T01:49:47Z","cross_cats_sorted":[],"title_canon_sha256":"915d7a00fbb4e119335e5c3ad580bdba13df653db000640331b45ed3c494c06f","abstract_canon_sha256":"f8ec300f86d53e31ad19a7ce989e19cd2cdb7106ea9f01ffcee80f37bbe3d025"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:18.789786Z","signature_b64":"TRR2WuXtdP599Vw/lGtMZ0Uxdrt88gjeCdVCPmhHkL7myspcMxqQ9S3OfYGLwRyW7XxvOyDEPNpNx+fGuXeiDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c218680863e61ba7511e6c140a1c94d9faae88179e7ffd09328322a17d1f17b","last_reissued_at":"2026-05-18T00:16:18.789290Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:18.789290Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$M_2$-Ranks of overpartitions modulo $6$ and $10$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Helen W.J. Zhang","submitted_at":"2018-05-10T01:49:47Z","abstract_excerpt":"In this paper, we obtain inequalities on $M_2$-ranks of overpartitions modulo $6$. Let $\\overline{N}_2(s,m,n)$ to be the number of overpartitions of $n$ whose $M_2$-rank is congruent to $s$ modulo $m$. For $M_2$-ranks modulo $3$, Lovejoy and Osburn derived the generating function of $\\overline{N}_2(s,3,n)-\\overline{N}_2(t,3,n)$, which implies the inequalities $\\overline{N}_2(0,3,n)\\geq\\overline{N}_2(1,3,n)$. For $\\ell=6, 10$, we consider the generating function $\\overline{R}_{s,t}(d,\\ell)$ of the $M_2$-rank differences $\\overline{N}_2(s,\\ell,\\ell n/2+d) + \\overline{N}_2(s+1,\\ell,\\ell n/2+d) - "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.03780","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":""},"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":"1805.03780","created_at":"2026-05-18T00:16:18.789385+00:00"},{"alias_kind":"arxiv_version","alias_value":"1805.03780v1","created_at":"2026-05-18T00:16:18.789385+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.03780","created_at":"2026-05-18T00:16:18.789385+00:00"},{"alias_kind":"pith_short_12","alias_value":"HQQYNAEGHZQ3","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_16","alias_value":"HQQYNAEGHZQ3U5IR","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_8","alias_value":"HQQYNAEG","created_at":"2026-05-18T12:32:28.185984+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/HQQYNAEGHZQ3U5IR43AUBIOJJW","json":"https://pith.science/pith/HQQYNAEGHZQ3U5IR43AUBIOJJW.json","graph_json":"https://pith.science/api/pith-number/HQQYNAEGHZQ3U5IR43AUBIOJJW/graph.json","events_json":"https://pith.science/api/pith-number/HQQYNAEGHZQ3U5IR43AUBIOJJW/events.json","paper":"https://pith.science/paper/HQQYNAEG"},"agent_actions":{"view_html":"https://pith.science/pith/HQQYNAEGHZQ3U5IR43AUBIOJJW","download_json":"https://pith.science/pith/HQQYNAEGHZQ3U5IR43AUBIOJJW.json","view_paper":"https://pith.science/paper/HQQYNAEG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1805.03780&json=true","fetch_graph":"https://pith.science/api/pith-number/HQQYNAEGHZQ3U5IR43AUBIOJJW/graph.json","fetch_events":"https://pith.science/api/pith-number/HQQYNAEGHZQ3U5IR43AUBIOJJW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HQQYNAEGHZQ3U5IR43AUBIOJJW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HQQYNAEGHZQ3U5IR43AUBIOJJW/action/storage_attestation","attest_author":"https://pith.science/pith/HQQYNAEGHZQ3U5IR43AUBIOJJW/action/author_attestation","sign_citation":"https://pith.science/pith/HQQYNAEGHZQ3U5IR43AUBIOJJW/action/citation_signature","submit_replication":"https://pith.science/pith/HQQYNAEGHZQ3U5IR43AUBIOJJW/action/replication_record"}},"created_at":"2026-05-18T00:16:18.789385+00:00","updated_at":"2026-05-18T00:16:18.789385+00:00"}