{"paper":{"title":"Proofs and generalizations of a homomesy conjecture of Propp and Roby","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dan Saracino, Jonathan Bloom, Oliver Pechenik","submitted_at":"2013-08-02T15:54:42Z","abstract_excerpt":"Let $G$ be a group acting on a set $X$ of combinatorial objects, with finite orbits, and consider a statistic $\\xi : X \\to \\mathbb{C}$. Propp and Roby defined the triple $(X, G, \\xi)$ to be \\emph{homomesic} if for any orbits $\\mathcal{O}_1, \\mathcal{O}_2$, the average value of the statistic $\\xi$ is the same, that is \\[\\frac{1}{{|\\mathcal{O}_1|}}\\sum_{x \\in \\mathcal{O}_1} \\xi(x) = \\frac{1}{|\\mathcal{O}_2|}\\sum_{y \\in \\mathcal{O}_2} \\xi(y).\\]\n  In 2013 Propp and Roby conjectured the following instance of homomesy. Let $\\mathrm{SSYT}_k(m \\times n)$ denote the set of semistandard Young tableaux o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.0546","kind":"arxiv","version":3},"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"}