{"paper":{"title":"The Duval--Reiner Conjecture: Counterexamples and the Second Partial-Sum Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jing Huang","submitted_at":"2026-07-22T11:49:13Z","abstract_excerpt":"Let \\(F\\subseteq\\binom{V}{q}\\) be a \\(q\\)-uniform family on a finite vertex set \\(V\\). Write \\(s_r(F)\\) for the sum of the \\(r\\) largest eigenvalues of its simplicial up-Laplacian and \\(d_F(v)\\) for the degree of \\(v\\in V\\). Then $D_r(F)=\\sum_{v\\in V}\\min\\{d_F(v),r\\}$ is the \\(r\\)-th partial sum of the conjugate degree sequence of \\(F\\). The majorization assertion in the Duval--Reiner conjecture [Trans. Amer. Math. Soc., 2002] states that \\(s_r(F)\\le D_r(F)\\) for every \\(q\\)-uniform family \\(F\\) and every \\(r\\ge1\\). We disprove this assertion in two complementary senses: for every \\(r\\ge5\\), t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20051","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/2607.20051/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"}