{"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"}