{"paper":{"title":"Arithmetic Properties of Partition Quadruples With Odd Parts Distinct","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2014-10-21T12:05:22Z","abstract_excerpt":"Let $\\mathrm{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\\mathrm{pod}_{-4}(n)$ involving the following infinite family of congruences: for any integers $\\alpha \\ge 1$ and $n \\ge 0$,\n  \\[\\mathrm{pod}_{-4}\\Big({{3}^{\\alpha +1}}n+\\frac{5\\cdot {{3}^{\\alpha }}+1}{2}\\Big)\\equiv 0 \\pmod{9}.\\] We also establish some internal congruences and some congruences modulo 2, 5 and 8 satisfied by $\\mathrm{pod}_{-4}(n)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5628","kind":"arxiv","version":2},"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"}