{"paper":{"title":"A combinatorial proof for the positivity of the normalized Jacobi triple product tails","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lisa Hui Sun, Xiangyu Ding","submitted_at":"2026-06-25T19:39:19Z","abstract_excerpt":"For $k\\geq 1$, we prove that \\[ [q^n z^s]J_k(z,q)\\geq 0, \\qquad (n\\geq 0,\\ s\\in\\mathbb Z) \\] for the normalized Jacobi triple product tails \\[ J_k(z,q) = \\frac{ \\sum_{j=k}^{\\infty}(-1)^{j-k} q^{\\binom{j+1}{2}}(z^{-j}+\\cdots+z^j)} {(zq,q/z;q)_\\infty}. \\] This result not only implies Merca's stronger nonnegativity conjecture on truncated Jacobi triple product series in full generality, but also yields infinite families of linear inequalities for two-colored partitions and partitions with parts in the residue classes $\\pm S \\pmod{R}$. We present a combinatorial proof wherein a sign-reversing invo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27507","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/2606.27507/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"}