{"title":"A generalization of a combinatorial theorem of macaulay","authors":"G.F. Clements , B. Lindström","doi":"10.1016/S0021-9800(69)80016-5","DOIUrl":"10.1016/S0021-9800(69)80016-5","url":null,"abstract":"<div><p>Let <em>E</em> denote the set of all vectors of dimension <em>n</em> (<em>n</em>≥2) with non-negative integral components. <em>E</em> is ordered in the lexicographic order. Let <em>E<sub>v</sub></em> denote the subset of all vectors in <em>E</em> with component sum <em>v</em>. If <em>H<sub>v</sub></em> denotes any subset of <em>E<sub>v</sub></em> let <em>LH<sub>v</sub></em> denote the set of the →<em>H<sub>v</sub></em>→ last elements in <em>E<sub>v</sub></em>, where →<em>H<sub>v</sub></em>→ is the number of elements of <em>H<sub>v</sub></em>. Let <em>PH<sub>v</sub></em> denote the set of all vectors of <em>E</em><sub><em>v</em>+1</sub>, which are obtained by the addition of 1 to a component of a vector in <em>H<sub>v</sub></em>. In [3] Macaulay proved the inclusion <em>P</em>(<em>LH<sub>v</sub></em>) ⊂ <em>L</em>(<em>PH<sub>v</sub></em>). Sperner gave a shorter proof in [4]. Let <em>k</em><sub>1</sub> ≤ <em>k</em><sub>2</sub> ≤ … ≤ <em>k<sub>n</sub></em> be given positive integers and let <em>F</em> denote the set of all vectors (<em>a</em><sub>1</sub>,…,<em>a<sub>n</sub></em>) with integer components and 0 ≤ <em>a<sub>i</sub></em> ≤ <em>k<sub>i</sub> i</em> = 1,…, <em>n</em>. We shall prove Macaulay's inclusion for subsets <em>H<sub>v</sub></em> of <em>F<sub>v</sub></em> even if the operators <em>P</em> and <em>L</em> are restricted to operate in <em>F</em>. This will follow from our theorem. As another application we prove a generalization of the main result in [2]. By a different method Katona proved the theorem when <em>k</em><sub>1</sub> = <em>k</em><sub>2</sub> = … = <em>k<sub>n</sub></em> = 1 (see [1, Theorem 1]).</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 3","pages":"Pages 230-238"},"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)80016-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90272670","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 furnishing Room squares","authors":"R.C. Mullin , E. Nemeth","doi":"10.1016/S0021-9800(69)80022-0","DOIUrl":"10.1016/S0021-9800(69)80022-0","url":null,"abstract":"<div><p>The authors show that for <em>v</em>=6<em>t</em>+1=<em>p<sup>n</sup>, p</em> a prime, there exists a Room square of order <em>v</em>+1.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 3","pages":"Pages 266-272"},"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)80022-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77109504","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":"A new proof of Brooks's chromatic number theorem for graphs","authors":"J. Ponstein","doi":"10.1016/S0021-9800(69)80018-9","DOIUrl":"10.1016/S0021-9800(69)80018-9","url":null,"abstract":"<div><p>The main purpose of this paper is to present one more proof of a theorem due to Brooks [1, 2, 3].</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 3","pages":"Pages 255-257"},"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)80018-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79037622","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":"Some new partition theorems. II","authors":"George E. Andrews","doi":"10.1016/S0021-9800(69)80020-7","DOIUrl":"https://doi.org/10.1016/S0021-9800(69)80020-7","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 3","pages":"Pages 262-263"},"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)80020-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"137216932","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":"Packing a rectangular box with the twelve solid pentominoes","authors":"C.J. Bouwkamp","doi":"10.1016/S0021-9800(69)80025-6","DOIUrl":"10.1016/S0021-9800(69)80025-6","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 3","pages":"Pages 278-280"},"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)80025-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87466802","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":"A class of covers for finite projective geometries which are related to the design of combinatorial filing systems","authors":"Gary G. Koch","doi":"10.1016/S0021-9800(69)80014-1","DOIUrl":"10.1016/S0021-9800(69)80014-1","url":null,"abstract":"<div><p>A method is given for selecting a certain subset of <em>m</em>-flats from a finite projective geometry <em>PG(N, q)</em> which cover all (<em>t</em>−1) flats there, where <em>m</em>≥<em>t</em>−1. These results have application to the problem of designing efficient information retrieval systems.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 3","pages":"Pages 215-220"},"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)80014-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78886383","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":"Some inequalities from switching theory","authors":"J.L. Brenner","doi":"10.1016/S0021-9800(69)80012-8","DOIUrl":"10.1016/S0021-9800(69)80012-8","url":null,"abstract":"<div><p><em>k</em>-Dimensional arrays of multiway switches are used to derive inequalities. (Analytic proofs are also given.) As an example, suppose 0<<em>p, q, r, …; p+q+r+ …</em>=1; <em>l, m, n, …>1, S=lmn …</em>. Interpreting <em>p, q, r</em>, … as probabilities that an individual switch be set in a coordinate direction, we derive</p><p><em>k</em>−1<(1−<em>p<sup>S/l</sup></em>)<sup><em>l</em></sup>+(1−<em>q<sup>S/m</sup></em>)<sup><em>m</em></sup>+(1−<em>r<sup>S/n</sup></em>)<sup><em>n</em></sup>+ …</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"7 3","pages":"Pages 197-205"},"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)80012-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86746408","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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