Pub Date : 1969-12-01DOI: 10.1016/S0021-9800(69)80058-X
N. Metropolis, M.L. Stein, P.R. Stein
An efficient method is presented for evaluating the permanents Pnk of cyclic (0,1) matrices of dimension n and common row and column sum k. A general method is developed for finding recurrence rules for Pnk (k fixed); the recurrence rules are given in semiexplicit form for the range 4≤k≤9. A table of Pnk is included for the range 4≤k≤9, k≤n≤80. The Pnk are calculated in the formwhere the Ttk(n) satisfy recurrence rules given symbolically by the characteristic equations of certain (0, 1) matrices Πrk; the latter turn out to be identical with the r-th permanental compounds of certain simpler matrices Π1k. Finally, formal expressions for Pnk 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.
{"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}
Pub Date : 1969-12-01DOI: 10.1016/S0021-9800(69)80057-8
L.S. Melnikov, V.G. Vizing
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.
新证明比原证明短，强调了双色链重着色在图的色数问题中的重要作用。
{"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}
Pub Date : 1969-12-01DOI: 10.1016/S0021-9800(69)80060-8
George K. Francis
The plane realizability problem for an abstract signed intersection sequence is solved by a twelve-instruction finite-decision algorithm.
用一种十二指令有限决策算法解决了抽象符号交序列的平面可实现性问题。
{"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}
Pub Date : 1969-12-01DOI: 10.1016/S0021-9800(69)80068-2
{"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}
Pub Date : 1969-12-01DOI: 10.1016/S0021-9800(69)80069-4
{"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}
Pub Date : 1969-11-01DOI: 10.1016/S0021-9800(69)80011-6
T.J. Dickson
In this paper we define a completely separating system of an n-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 n, is asymptotic to the cardinality of a minimal separating system.
{"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}
Pub Date : 1969-11-01DOI: 10.1016/S0021-9800(69)80013-X
Robert J. Douglas
{"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}
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