{"title":"Permanents of cyclic (0,1) matrices","authors":"N. Metropolis, M.L. Stein, P.R. Stein","doi":"10.1016/S0021-9800(69)80058-X","DOIUrl":"https://doi.org/10.1016/S0021-9800(69)80058-X","url":null,"abstract":"<div><p>An efficient method is presented for evaluating the permanents <em>P<sub>n</sub><sup>k</sup></em> of cyclic (0,1) matrices of dimension <em>n</em> and common row and column sum <em>k</em>. A general method is developed for finding recurrence rules for <em>P<sub>n</sub><sup>k</sup></em> (<em>k</em> fixed); the recurrence rules are given in semiexplicit form for the range 4≤<em>k</em>≤9. A table of <em>P<sub>n</sub><sup>k</sup></em> is included for the range 4≤<em>k</em>≤9, <em>k</em>≤<em>n</em>≤80. The <em>P<sub>n</sub><sup>k</sup></em> are calculated in the form<span><span><span><math><mrow><msubsup><mi>P</mi><mi>n</mi><mo>k</mo></msubsup><mo>=</mo><mn>2</mn><mo>+</mo><mstyle><munderover><mo>∑</mo><mrow><mi>τ</mi><mo>−</mo><mn>1</mn></mrow><mrow><mo>[</mo><mstyle><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mstyle><mo>]</mo></mrow></munderover><mrow><msubsup><mi>T</mi><mi>τ</mi><mo>k</mo></msubsup><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mstyle></mrow></math></span></span></span>where the <em>T<sub>t</sub><sup>k</sup>(n)</em> satisfy recurrence rules given symbolically by the characteristic equations of certain (0, 1) matrices <em>Π<sub>r</sub><sup>k</sup></em>; the latter turn out to be identical with the <em>r</em>-th permanental compounds of certain simpler matrices <em>Π<sub>1</sub><sup>k</sup></em>. Finally, formal expressions for <em>P<sub>n</sub><sup>k</sup></em> are given which allow one to write down the solution to the generalized Ménage Problem in terms of sums over scalar products of the iterates of a set of unit vectors.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 4","pages":"Pages 291-321"},"PeriodicalIF":0.0,"publicationDate":"1969-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(69)80058-X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91637968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"New proof of brooks' theorem","authors":"L.S. Melnikov, V.G. Vizing","doi":"10.1016/S0021-9800(69)80057-8","DOIUrl":"https://doi.org/10.1016/S0021-9800(69)80057-8","url":null,"abstract":"<div><p>The new proof is shorter than the original one [1] and emphasizes the important role of recoloring of two-color chains in questions related to chromatic number of graphs.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 4","pages":"Pages 289-290"},"PeriodicalIF":0.0,"publicationDate":"1969-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(69)80057-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91637970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Null genus realizability criterion for abstract intersection sequences","authors":"George K. Francis","doi":"10.1016/S0021-9800(69)80060-8","DOIUrl":"https://doi.org/10.1016/S0021-9800(69)80060-8","url":null,"abstract":"<div><p>The plane realizability problem for an abstract signed intersection sequence is solved by a twelve-instruction finite-decision algorithm.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 4","pages":"Pages 331-341"},"PeriodicalIF":0.0,"publicationDate":"1969-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(69)80060-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91610973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Addendum to previously published paper","authors":"","doi":"10.1016/S0021-9800(69)80068-2","DOIUrl":"https://doi.org/10.1016/S0021-9800(69)80068-2","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 4","pages":"Page 378"},"PeriodicalIF":0.0,"publicationDate":"1969-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(69)80068-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91636675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Author index of volume 7","authors":"","doi":"10.1016/S0021-9800(69)80069-4","DOIUrl":"https://doi.org/10.1016/S0021-9800(69)80069-4","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 4","pages":"Pages 379-380"},"PeriodicalIF":0.0,"publicationDate":"1969-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(69)80069-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136907189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a problem concerning separating systems of a finite set","authors":"T.J. Dickson","doi":"10.1016/S0021-9800(69)80011-6","DOIUrl":"10.1016/S0021-9800(69)80011-6","url":null,"abstract":"<div><p>In this paper we define a completely separating system of an <em>n</em>-set, an extension of the concept of a separating system introduced by Renyi [1] for use in certain information theoretic problems. We then consider the problem of finding the cardinality of a minimal completely separating system and show that this, considered as a function of <em>n</em>, is asymptotic to the cardinality of a minimal separating system.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 3","pages":"Pages 191-196"},"PeriodicalIF":0.0,"publicationDate":"1969-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(69)80011-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75203961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Upper bounds on the length of circuits of even spread in the d-cube","authors":"Robert J. Douglas","doi":"10.1016/S0021-9800(69)80013-X","DOIUrl":"10.1016/S0021-9800(69)80013-X","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 3","pages":"Pages 206-214"},"PeriodicalIF":0.0,"publicationDate":"1969-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(69)80013-X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79399360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
求助内容：
应助结果提醒方式：