European Journal of Combinatorics最新文献

{"title":"An identity of Ramanujan and its combinatorics","authors":"Bernard L.S. Lin ,&nbsp;Xiaowei Lin ,&nbsp;Lei Zhang","doi":"10.1016/j.ejc.2024.103985","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103985","url":null,"abstract":"<div><p>In this paper, we explore the combinatorics behind an identity recorded in Ramanujan’s lost notebook. We present an interesting result, which not only generalizes two theorems of Bressoud, but also implies a bivariate form of Ramanujan’s original identity.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103985"},"PeriodicalIF":1.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140902018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bounds for the number of multidimensional partitions","authors":"Kristina Oganesyan","doi":"10.1016/j.ejc.2024.103982","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103982","url":null,"abstract":"<div><p>We obtain estimates for the number <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-dimensional integer partitions of a number <span><math><mi>n</mi></math></span>. It is known that the two-sided inequality <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>d</mi></mrow></msup><mo>&lt;</mo><mo>log</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>&lt;</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>d</mi></mrow></msup></mrow></math></span> is always true and that <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow><mo>&gt;</mo><mn>1</mn></mrow></math></span> whenever <span><math><mrow><mo>log</mo><mi>n</mi><mo>&gt;</mo><mn>3</mn><mi>d</mi></mrow></math></span>. However, establishing the <span><math><mi>“</mi></math></span>right<span><math><mi>”</mi></math></span> dependence of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> on <span><math><mi>d</mi></math></span> remained an open problem. We show that if <span><math><mi>d</mi></math></span> is sufficiently small with respect to <span><math><mi>n</mi></math></span>, then <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> does not depend on <span><math><mi>d</mi></math></span>, which means that <span><math><mrow><mo>log</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> is up to an absolute constant equal to <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>d</mi></mrow></msup></math></span>. Besides, we provide estimates of <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> for different ranges of <span><math><mi>d</mi></math></span> in terms of <span><math><mi>n</mi></math></span>, which give the asymptotics of <span><math><mrow><mo>log</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> in each case.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103982"},"PeriodicalIF":1.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140902016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two Ramsey problems in blowups of graphs","authors":"António Girão ,&nbsp;Robert Hancock","doi":"10.1016/j.ejc.2024.103984","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103984","url":null,"abstract":"<div><p>Given graphs <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span>, we say <span><math><mrow><mi>G</mi><mover><mrow><mo>→</mo></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mover><mi>H</mi></mrow></math></span> if every <span><math><mi>r</mi></math></span>-colouring of the edges of <span><math><mi>G</mi></math></span> contains a monochromatic copy of <span><math><mi>H</mi></math></span>. Let <span><math><mrow><mi>H</mi><mrow><mo>[</mo><mi>t</mi><mo>]</mo></mrow></mrow></math></span> denote the <span><math><mi>t</mi></math></span>-blowup of <span><math><mi>H</mi></math></span>. The blowup Ramsey number <span><math><mrow><mi>B</mi><mrow><mo>(</mo><mi>G</mi><mover><mrow><mo>→</mo></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mover><mi>H</mi><mo>;</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> is the minimum <span><math><mi>n</mi></math></span> such that <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mover><mrow><mo>→</mo></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mover><mi>H</mi><mrow><mo>[</mo><mi>t</mi><mo>]</mo></mrow></mrow></math></span>. Fox, Luo and Wigderson refined an upper bound of Souza, showing that, given <span><math><mi>G</mi></math></span>, <span><math><mi>H</mi></math></span> and <span><math><mi>r</mi></math></span> such that <span><math><mrow><mi>G</mi><mover><mrow><mo>→</mo></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mover><mi>H</mi></mrow></math></span>, there exist constants <span><math><mrow><mi>a</mi><mo>=</mo><mi>a</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>b</mi><mo>=</mo><mi>b</mi><mrow><mo>(</mo><mi>H</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> such that for all <span><math><mrow><mi>t</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, <span><math><mrow><mi>B</mi><mrow><mo>(</mo><mi>G</mi><mover><mrow><mo>→</mo></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mover><mi>H</mi><mo>;</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><mi>a</mi><msup><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></math></span>. They conjectured that there exist some graphs <span><math><mi>H</mi></math></span> for which the constant <span><math><mi>a</mi></math></span> depending on <span><math><mi>G</mi></math></span> is necessary. We prove this conjecture by showing that the statement is true in the case of <span><math><mi>H</mi></math></span> being 3-chromatically connected, which in particular includes triangles. On the other hand, perhaps surprisingly, we show that for forests <span><math><mi>F</mi></math></span>, there exists an upper bound for <span><math><mrow><mi>B</mi><mrow><mo>(</mo><mi>G</mi><mover><mrow><mo>→</mo></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mover><mi>F</mi><mo>;</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> which is independent of <span><math><mi>G</mi></math></span>.</p><p>Second, we show that for any <span><math><mrow><mi>r</mi><mo>,</mo><mi","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103984"},"PeriodicalIF":1.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000696/pdfft?md5=90c712e911fae05fd1803c79c5bbceb8&pid=1-s2.0-S0195669824000696-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140902017","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Monochromatic triangles in the max-norm plane","authors":"Alexander Natalchenko ,&nbsp;Arsenii Sagdeev","doi":"10.1016/j.ejc.2024.103977","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103977","url":null,"abstract":"<div><p>For all non-degenerate triangles <span><math><mi>T</mi></math></span>, we determine the minimum number of colors needed to color the plane such that no max-norm isometric copy of <span><math><mi>T</mi></math></span> is monochromatic.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103977"},"PeriodicalIF":1.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140843589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cycles in Austrian Solitaire","authors":"Philip P. Mummert","doi":"10.1016/j.ejc.2024.103978","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103978","url":null,"abstract":"<div><p>Austrian Solitaire is a variation of Bulgarian Solitaire. It may be described as a card game, a method of asset inventory management, or a discrete dynamical system on integer partitions. We prove that the limit cycles in Austrian Solitaire do not depend on the initial configuration; in other words, each state space is connected. We show that a full Farey sequence completely characterizes these unique (and balanced) cycles.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103978"},"PeriodicalIF":1.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140842747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Arc-disjoint out- and in-branchings in compositions of digraphs","authors":"J. Bang-Jensen ,&nbsp;Y. Wang","doi":"10.1016/j.ejc.2024.103981","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103981","url":null,"abstract":"<div><p>An out-branching <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> (in-branching <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>u</mi></mrow><mrow><mo>−</mo></mrow></msubsup></math></span>) in a digraph <span><math><mi>D</mi></math></span> is a connected spanning subdigraph of <span><math><mi>D</mi></math></span> in which every vertex except the vertex <span><math><mi>u</mi></math></span>, called the root, has in-degree (out-degree) one. A <strong>good</strong><span><math><mi>(u,v)</mi></math></span>-<strong>pair</strong> in <span><math><mi>D</mi></math></span> is a pair of branchings <span><math><mrow><msubsup><mrow><mi>B</mi></mrow><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>v</mi></mrow><mrow><mo>−</mo></mrow></msubsup></mrow></math></span> which have no arc in common. Thomassen proved that it is NP-complete to decide if a digraph has any good pair. A digraph is <strong>semicomplete</strong> if it has no pair of non-adjacent vertices. A <strong>semicomplete composition</strong> is any digraph <span><math><mi>D</mi></math></span> which is obtained from a semicomplete digraph <span><math><mi>S</mi></math></span> by substituting an arbitrary digraph <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span> for each vertex <span><math><mi>x</mi></math></span> of <span><math><mi>S</mi></math></span>.</p><p>Recently the authors of this paper gave a complete classification of semicomplete digraphs which have a good <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></math></span>-pair, where <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></math></span> are prescribed vertices. They also gave a polynomial algorithm which for a given semicomplete digraph <span><math><mi>D</mi></math></span> and vertices <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></math></span> of <span><math><mi>D</mi></math></span>, either produces a good <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></math></span>-pair in <span><math><mi>D</mi></math></span> or a certificate that <span><math><mi>D</mi></math></span> has no such pair. In this paper we show how to use the result for semicomplete digraphs to completely solve the problem of characterizing semicomplete compositions which have a good <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></math></span>-pair for given vertices <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></math></span>. Our solution implies that the problem of deciding the existence of a good <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></math></span>-pair and finding such a pair when it exists is polynomially solvable for all semicomplete compositions. We also completely solve the problem of deciding the existence of a good <span><math><mrow><mo>(</mo><mi>u","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103981"},"PeriodicalIF":1.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000660/pdfft?md5=9f52566484a640f1db27537236930da5&pid=1-s2.0-S0195669824000660-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140843408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Counting deranged matchings","authors":"Sam Spiro ,&nbsp;Erlang Surya","doi":"10.1016/j.ejc.2024.103980","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103980","url":null,"abstract":"<div><p>Let <span><math><mrow><mi>pm</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denote the number of perfect matchings of a graph <span><math><mi>G</mi></math></span>, and let <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>×</mo><mn>2</mn><mi>n</mi><mo>/</mo><mi>r</mi></mrow></msub></math></span> denote the complete <span><math><mi>r</mi></math></span>-partite graph where each part has size <span><math><mrow><mn>2</mn><mi>n</mi><mo>/</mo><mi>r</mi></mrow></math></span>. Johnson, Kayll, and Palmer conjectured that for any perfect matching <span><math><mi>M</mi></math></span> of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>×</mo><mn>2</mn><mi>n</mi><mo>/</mo><mi>r</mi></mrow></msub></math></span>, we have for <span><math><mrow><mn>2</mn><mi>n</mi></mrow></math></span> divisible by <span><math><mi>r</mi></math></span>\u0000<span><span><span><math><mrow><mfrac><mrow><mi>pm</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>×</mo><mn>2</mn><mi>n</mi><mo>/</mo><mi>r</mi></mrow></msub><mo>−</mo><mi>M</mi><mo>)</mo></mrow></mrow><mrow><mi>pm</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>×</mo><mn>2</mn><mi>n</mi><mo>/</mo><mi>r</mi></mrow></msub><mo>)</mo></mrow></mrow></mfrac><mo>∼</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>r</mi><mo>/</mo><mrow><mo>(</mo><mn>2</mn><mi>r</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow></msup><mo>.</mo></mrow></math></span></span></span>This conjecture can be viewed as a common generalization of counting the number of derangements on <span><math><mi>n</mi></math></span> letters, and of counting the number of deranged matchings of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span>. We prove this conjecture. In fact, we prove the stronger result that if <span><math><mi>R</mi></math></span> is a uniformly random perfect matching of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>×</mo><mn>2</mn><mi>n</mi><mo>/</mo><mi>r</mi></mrow></msub></math></span>, then the number of edges that <span><math><mi>R</mi></math></span> has in common with <span><math><mi>M</mi></math></span> converges to a Poisson distribution with parameter <span><math><mfrac><mrow><mi>r</mi></mrow><mrow><mn>2</mn><mi>r</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span>.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103980"},"PeriodicalIF":1.0,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000659/pdfft?md5=c66422b992cbbf0765bc7eba3abcde93&pid=1-s2.0-S0195669824000659-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140650508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Large Y3,2-tilings in 3-uniform hypergraphs","authors":"Jie Han ,&nbsp;Lin Sun ,&nbsp;Guanghui Wang","doi":"10.1016/j.ejc.2024.103976","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103976","url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>2</mn></mrow></msub></math></span> be the 3-graph with two edges intersecting in two vertices. We prove that every 3-graph <span><math><mi>H</mi></math></span> on <span><math><mi>n</mi></math></span> vertices with at least <span><math><mrow><mo>max</mo><mfenced><mrow><mfenced><mrow><mfrac><mrow><mn>4</mn><mi>α</mi><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced><mo>,</mo><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced><mo>−</mo><mfenced><mrow><mfrac><mrow><mi>n</mi><mo>−</mo><mi>α</mi><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>+</mo><mi>o</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> edges contains a <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>2</mn></mrow></msub></math></span>-tiling covering more than <span><math><mrow><mn>4</mn><mi>α</mi><mi>n</mi></mrow></math></span> vertices, for sufficiently large <span><math><mi>n</mi></math></span> and <span><math><mrow><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mn>1</mn><mo>/</mo><mn>4</mn></mrow></math></span>. The bound on the number of edges is asymptotically best possible and solves a conjecture of the authors for 3-graphs that generalizes the Matching Conjecture of Erdős.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103976"},"PeriodicalIF":1.0,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140807169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Jungerman ladders and index 2 constructions for genus embeddings of dense regular graphs","authors":"Timothy Sun","doi":"10.1016/j.ejc.2024.103974","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103974","url":null,"abstract":"<div><p>We construct several families of minimum genus embeddings of dense graphs using index 2 current graphs. In particular, we complete the genus formula for the octahedral graphs, solving a longstanding conjecture of Jungerman and Ringel, and find triangular embeddings of complete graphs minus a Hamiltonian cycle, making partial progress on a problem of White. Index 2 current graphs are also applied to various cases of the genus of the complete graphs, in some cases yielding simpler solutions, e.g., the nonorientable genus of <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>12</mn><mi>s</mi><mo>+</mo><mn>8</mn></mrow></msub><mo>−</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>. In addition, we give a simpler proof of a theorem of Jungerman that shows that a symmetric type of such current graphs might not exist roughly “half of the time”.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103974"},"PeriodicalIF":1.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000593/pdfft?md5=a4ea0dfe22653b0b9f17b72d23ecb25a&pid=1-s2.0-S0195669824000593-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140644394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Symmetry and Pieri rules for the bisymmetric Macdonald polynomials","authors":"Manuel Concha,&nbsp;Luc Lapointe","doi":"10.1016/j.ejc.2024.103973","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103973","url":null,"abstract":"<div><p>Bisymmetric Macdonald polynomials can be obtained through a process of antisymmetrization and <span><math><mi>t</mi></math></span>-symmetrization of non-symmetric Macdonald polynomials. Using the double affine Hecke algebra, we show that the evaluation of the bisymmetric Macdonald polynomials satisfies a symmetry property generalizing that satisfied by the usual Macdonald polynomials. We then obtain Pieri rules for the bisymmetric Macdonald polynomials where the sums are over certain vertical strips.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103973"},"PeriodicalIF":1.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140644393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}