{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:ZQPMN7KS5U5JEXDT5ORHL6AICT","short_pith_number":"pith:ZQPMN7KS","schema_version":"1.0","canonical_sha256":"cc1ec6fd52ed3a925c73eba275f80814dd22ca5d314fbd5adf0e58b646367bc3","source":{"kind":"arxiv","id":"1701.04321","version":2},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1701.04321","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-16T15:16:15Z","cross_cats_sorted":[],"title_canon_sha256":"c6725b8a1a56e1d0e33ae88cfc92455afea245996ba7e9991999d20ceeec6636","abstract_canon_sha256":"d16d0784975d4898916447442c7ff97807ead378fc23452b352b7719c17b2c01"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:59.473536Z","signature_b64":"o4H1YLmS2JiCavfuv7l3F21eyOiMtLBPEIl2fVeC09mOxR42+NJ/V6i9EZsfBmNSUb1I76cTar1DXSfJoDy4Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cc1ec6fd52ed3a925c73eba275f80814dd22ca5d314fbd5adf0e58b646367bc3","last_reissued_at":"2026-05-18T00:48:59.472757Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:59.472757Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1701.04321","created_at":"2026-05-18T00:48:59.472870+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.04321v2","created_at":"2026-05-18T00:48:59.472870+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.04321","created_at":"2026-05-18T00:48:59.472870+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZQPMN7KS5U5J","created_at":"2026-05-18T12:31:59.375834+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZQPMN7KS5U5JEXDT","created_at":"2026-05-18T12:31:59.375834+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZQPMN7KS","created_at":"2026-05-18T12:31:59.375834+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZQPMN7KS5U5JEXDT5ORHL6AICT","json":"https://pith.science/pith/ZQPMN7KS5U5JEXDT5ORHL6AICT.json","graph_json":"https://pith.science/api/pith-number/ZQPMN7KS5U5JEXDT5ORHL6AICT/graph.json","events_json":"https://pith.science/api/pith-number/ZQPMN7KS5U5JEXDT5ORHL6AICT/events.json","paper":"https://pith.science/paper/ZQPMN7KS"},"agent_actions":{"view_html":"https://pith.science/pith/ZQPMN7KS5U5JEXDT5ORHL6AICT","download_json":"https://pith.science/pith/ZQPMN7KS5U5JEXDT5ORHL6AICT.json","view_paper":"https://pith.science/paper/ZQPMN7KS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.04321&json=true","fetch_graph":"https://pith.science/api/pith-number/ZQPMN7KS5U5JEXDT5ORHL6AICT/graph.json","fetch_events":"https://pith.science/api/pith-number/ZQPMN7KS5U5JEXDT5ORHL6AICT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZQPMN7KS5U5JEXDT5ORHL6AICT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZQPMN7KS5U5JEXDT5ORHL6AICT/action/storage_attestation","attest_author":"https://pith.science/pith/ZQPMN7KS5U5JEXDT5ORHL6AICT/action/author_attestation","sign_citation":"https://pith.science/pith/ZQPMN7KS5U5JEXDT5ORHL6AICT/action/citation_signature","submit_replication":"https://pith.science/pith/ZQPMN7KS5U5JEXDT5ORHL6AICT/action/replication_record"}},"created_at":"2026-05-18T00:48:59.472870+00:00","updated_at":"2026-05-18T00:48:59.472870+00:00"}