{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T09:06:32Z","timestamp":1751879192035},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2002,4,25]],"date-time":"2002-04-25T00:00:00Z","timestamp":1019692800000},"content-version":"unspecified","delay-in-days":55,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2002,3]]},"abstract":"<jats:p>Let <jats:bold>T<\/jats:bold><jats:sup><jats:italic>n<\/jats:italic><\/jats:sup> be the complete binary tree of height <jats:italic>n<\/jats:italic> considered as the Hasse diagram \n\nof a poset with its root <jats:bold>1<\/jats:bold><jats:sub><jats:bold>n<\/jats:bold><\/jats:sub> as the maximum element. Define <jats:italic>A<\/jats:italic>(<jats:italic>n<\/jats:italic>; <jats:italic>T<\/jats:italic>) \n\n= [mid     ]{<jats:italic>S<\/jats:italic> \u2286 <jats:bold>T<\/jats:bold><jats:sup><jats:bold>n<\/jats:bold><\/jats:sup> : <jats:bold>1<\/jats:bold><jats:sub><jats:bold>n<\/jats:bold><\/jats:sub> \u2208 <jats:italic>S<\/jats:italic>, \n\n<jats:italic>S<\/jats:italic> \u2245 <jats:italic>T<\/jats:italic>}[mid     ], and <jats:italic>B<\/jats:italic>(<jats:italic>n<\/jats:italic>; <jats:italic>T<\/jats:italic>) = [mid     ]{<jats:italic>S<\/jats:italic> \u2286 \n\n<jats:bold>T<\/jats:bold><jats:sup><jats:bold>n<\/jats:bold><\/jats:sup> : <jats:bold>1<\/jats:bold><jats:sub><jats:bold>n<\/jats:bold><\/jats:sub> \u2209 <jats:italic>S<\/jats:italic>, <jats:italic>S<\/jats:italic> \u2245 <jats:italic>T<\/jats:italic>}[mid     ]. \n\nIn this note we prove that <jats:disp-formula id=\"S0963548301004977_ffm001\"><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"ffm001\" xlink:href=\"\" \/><\/jats:disp-formula> for \n\nany fixed <jats:italic>n<\/jats:italic> and rooted binary trees <jats:italic>T<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>T<\/jats:italic><jats:sub>2<\/jats:sub> such that \n\n<jats:italic>T<\/jats:italic><jats:sub>2<\/jats:sub> contains a subposet isomorphic to <jats:italic>T<\/jats:italic><jats:sub>1<\/jats:sub>. We conjecture that the ratio \n\n<jats:italic>A<\/jats:italic>\/<jats:italic>B<\/jats:italic> also increases with <jats:italic>T<\/jats:italic> for arbitrary trees. These \n\ninequalities imply natural behaviour of the optimal stopping time in a poset extension of \n\nthe secretary problem.<\/jats:p>","DOI":"10.1017\/s0963548301004977","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T09:49:03Z","timestamp":1027763343000},"page":"149-161","source":"Crossref","is-referenced-by-count":5,"title":["A Ratio Inequality for Binary Trees and the Best Secretary"],"prefix":"10.1017","volume":"11","author":[{"given":"GRZEGORZ","family":"KUBICKI","sequence":"first","affiliation":[]},{"given":"JEN\u0150","family":"LEHEL","sequence":"additional","affiliation":[]},{"given":"MICHA\u0141","family":"MORAYNE","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2002,4,25]]},"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548301004977","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,3]],"date-time":"2019-04-03T15:53:51Z","timestamp":1554306831000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548301004977\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,3]]},"references-count":0,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2002,3]]}},"alternative-id":["S0963548301004977"],"URL":"https:\/\/doi.org\/10.1017\/s0963548301004977","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,3]]}}}