{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:28:24Z","timestamp":1787239704767,"version":"build-2736575974"},"reference-count":4,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM. J. on Algebraic and Discrete Methods"],"published-print":{"date-parts":[[1980,9]]},"abstract":"<jats:p>A fundamental quantity which arises in the sorting of n numbers $a_{1}, a_{2}, \\cdots ,a_n $ is $\\Pr ( a_i &lt; a_j |P )$, the probability that $a_i &lt; a_j $ assuming that all linear extensions of the partial order P are equally likely. In this paper, we establish various properties of $\\Pr ( a_i &lt; a_j |P )$ and related quantities. In particular, it is shown that $\\Pr ( a_i &lt; b_j |P^\\prime )\\geqq \\Pr ( a_i &lt; b_j |P )$ if the partial order P consists of two disjoint linearly ordered sets $A = \\{ a_1 &lt; a_2 &lt; \\cdots &lt; a_m \\},B = \\{ b_1 &lt; b_2 &lt; \\cdots &lt; b_n \\}$ and $P^\\prime = P \\cup \\{ \\text{any relations of the form} \\, a_k &lt; b_l\\}$. These inequalities have applications in determining the complexity of certain sorting-like computations.<\/jats:p>","DOI":"10.1137\/0601028","type":"journal-article","created":{"date-parts":[[2005,2,27]],"date-time":"2005-02-27T07:03:28Z","timestamp":1109487808000},"page":"251-258","source":"Crossref","is-referenced-by-count":33,"title":["Some Monotonicity Properties of Partial Orders"],"prefix":"10.1137","volume":"1","author":[{"given":"R. L.","family":"Graham","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A. C.","family":"Yao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"F. F.","family":"Yao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-10.37.26"},{"key":"R2","volume-title":"The art of computer programming. Volume 3","author":"Knuth Donald E.","year":"1973"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-1969-081-4"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1109\/SFCS.1978.29"}],"container-title":["SIAM Journal on Algebraic Discrete Methods"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/0601028","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:46:26Z","timestamp":1787237186000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/0601028"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1980,9]]},"references-count":4,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1980,9]]}},"alternative-id":["10.1137\/0601028"],"URL":"https:\/\/doi.org\/10.1137\/0601028","relation":{},"ISSN":["0196-5212","2168-345X"],"issn-type":[{"value":"0196-5212","type":"print"},{"value":"2168-345X","type":"electronic"}],"subject":[],"published":{"date-parts":[[1980,9]]}}}