{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:HQQYNAEGHZQ3U5IR43AUBIOJJW","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f8ec300f86d53e31ad19a7ce989e19cd2cdb7106ea9f01ffcee80f37bbe3d025","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-10T01:49:47Z","title_canon_sha256":"915d7a00fbb4e119335e5c3ad580bdba13df653db000640331b45ed3c494c06f"},"schema_version":"1.0","source":{"id":"1805.03780","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.03780","created_at":"2026-05-18T00:16:18Z"},{"alias_kind":"arxiv_version","alias_value":"1805.03780v1","created_at":"2026-05-18T00:16:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.03780","created_at":"2026-05-18T00:16:18Z"},{"alias_kind":"pith_short_12","alias_value":"HQQYNAEGHZQ3","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"HQQYNAEGHZQ3U5IR","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"HQQYNAEG","created_at":"2026-05-18T12:32:28Z"}],"graph_snapshots":[{"event_id":"sha256:10f2ce8d431f69f0811d5cbbe8382767a88df0aa97ce61527980bca7b8a8123c","target":"graph","created_at":"2026-05-18T00:16:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 $\\overline{N}_2(s,\\ell,\\ell n/2+d) + \\overline{N}_2(s+1,\\ell,\\ell n/2+d) - ","authors_text":"Helen W.J. Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-10T01:49:47Z","title":"$M_2$-Ranks of overpartitions modulo $6$ and $10$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.03780","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6260ecd4a59d15d14551e4f30b8b2afe805fa1bcadd2c665399d19421e6c7e4d","target":"record","created_at":"2026-05-18T00:16:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f8ec300f86d53e31ad19a7ce989e19cd2cdb7106ea9f01ffcee80f37bbe3d025","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-10T01:49:47Z","title_canon_sha256":"915d7a00fbb4e119335e5c3ad580bdba13df653db000640331b45ed3c494c06f"},"schema_version":"1.0","source":{"id":"1805.03780","kind":"arxiv","version":1}},"canonical_sha256":"3c218680863e61ba7511e6c140a1c94d9faae88179e7ffd09328322a17d1f17b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3c218680863e61ba7511e6c140a1c94d9faae88179e7ffd09328322a17d1f17b","first_computed_at":"2026-05-18T00:16:18.789290Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:18.789290Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TRR2WuXtdP599Vw/lGtMZ0Uxdrt88gjeCdVCPmhHkL7myspcMxqQ9S3OfYGLwRyW7XxvOyDEPNpNx+fGuXeiDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:18.789786Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.03780","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6260ecd4a59d15d14551e4f30b8b2afe805fa1bcadd2c665399d19421e6c7e4d","sha256:10f2ce8d431f69f0811d5cbbe8382767a88df0aa97ce61527980bca7b8a8123c"],"state_sha256":"082bb699e11001369a1c8548176405bc395d81cd354bee164b107d2ba89743eb"}