{"paper":{"title":"Proof of an entropy conjecture of Leighton and Moitra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"H\\\"useyin Acan, Jeff Kahn, Pat Devlin","submitted_at":"2017-01-16T15:16:15Z","abstract_excerpt":"We prove the following conjecture of Leighton and Moitra. Let $T$ be a tournament on $[n]$ and $S_n$ the set of permutations of $[n]$. For an arc $uv$ of $T$, let $A_{uv}=\\{\\sigma \\in S_n \\, : \\, \\sigma(u)<\\sigma(v) \\}$.\n  $\\textbf{Theorem.}$ For a fixed $\\varepsilon>0$, if $\\mathbb{P}$ is a probability distribution on $S_n$ such that $\\mathbb{P}(A_{uv})>1/2+\\varepsilon$ for every arc $uv$ of $T$, then the binary entropy of $\\mathbb{P}$ is at most $(1-\\vartheta_{\\varepsilon})\\log_2 n!$ for some (fixed) positive $\\vartheta_\\varepsilon$.\n  When $T$ is transitive the theorem is due to Leighton an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04321","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"}