{"paper":{"title":"Square-root cancellation for the signs of Latin squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Levent Alpoge","submitted_at":"2014-12-23T23:46:51Z","abstract_excerpt":"Let $L(n)$ be the number of Latin squares of order $n$, and let $L^{\\textrm{even}}(n)$ and $L^{\\textrm{odd}}(n)$ be the number of even and odd such squares, so that $L(n) = L^{\\textrm{even}}(n) + L^{\\textrm{odd}}(n)$. The Alon-Tarsi conjecture states that $L^{\\textrm{even}}(n)\\neq L^{\\textrm{odd}}(n)$ when $n$ is even (when $n$ is odd the two are equal for very simple reasons). In this short note we prove that $|L^{\\textrm{even}}(n) - L^{\\textrm{odd}}(n)|\\leq L(n)^{\\frac{1}{2} + o(1)},$ thus establishing the conjecture that the number of even and odd Latin squares, while conjecturally not equa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.7574","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"}