Journal of Combinatorial Theory Series A最新文献

{"title":"Point-line geometries related to binary equidistant codes","authors":"Mark Pankov ,&nbsp;Krzysztof Petelczyc ,&nbsp;Mariusz Żynel","doi":"10.1016/j.jcta.2024.105962","DOIUrl":"10.1016/j.jcta.2024.105962","url":null,"abstract":"<div><div>Point-line geometries whose singular subspaces correspond to binary equidistant codes are investigated. The main result is a description of automorphisms of these geometries. In some important cases, automorphisms induced by non-monomial linear automorphisms surprisingly arise.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105962"},"PeriodicalIF":0.9,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Neighborly partitions, hypergraphs and Gordon's identities","authors":"Pooneh Afsharijoo ,&nbsp;Hussein Mourtada","doi":"10.1016/j.jcta.2024.105963","DOIUrl":"10.1016/j.jcta.2024.105963","url":null,"abstract":"<div><div>We prove a family of partition identities which is “dual” to the family of Andrews-Gordon's identities. These identities are inspired by a correspondence between a special type of partitions and “hypergraphs” and their proof uses combinatorial commutative algebra.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105963"},"PeriodicalIF":0.9,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On locally n × n grid graphs","authors":"Carmen Amarra ,&nbsp;Wei Jin ,&nbsp;Cheryl E. Praeger","doi":"10.1016/j.jcta.2024.105957","DOIUrl":"10.1016/j.jcta.2024.105957","url":null,"abstract":"<div><div>We investigate locally <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> grid graphs, that is, graphs in which the neighbourhood of any vertex is the Cartesian product of two complete graphs on <em>n</em> vertices. We consider the subclass of these graphs for which each pair of vertices at distance two is joined by sufficiently many paths of length 2. The number of such paths is known to be at most 2<em>n</em> by previous work of Blokhuis and Brouwer. We show that if each pair is joined by at least <span><math><mn>2</mn><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> such paths then the diameter is at most 3 and we give a tight upper bound on the order of the graphs. We show that graphs meeting this upper bound are distance-regular antipodal covers of complete graphs. We exhibit an infinite family of such graphs which are locally <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> grid for odd prime powers <em>n</em>, and apply these results to locally <span><math><mn>5</mn><mo>×</mo><mn>5</mn></math></span> grid graphs to obtain a classification for the case where either all <em>μ</em>-graphs (induced subgraphs on the set of common neighbours of two vertices at distance two) have order at least 8 or all <em>μ</em>-graphs have order <em>c</em> for some constant <em>c</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105957"},"PeriodicalIF":0.9,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142322889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On power monoids and their automorphisms","authors":"Salvatore Tringali,&nbsp;Weihao Yan","doi":"10.1016/j.jcta.2024.105961","DOIUrl":"10.1016/j.jcta.2024.105961","url":null,"abstract":"<div><div>Endowed with the binary operation of set addition, the family <span><math><msub><mrow><mi>P</mi></mrow><mrow><mrow><mi>fin</mi></mrow><mo>,</mo><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> of all finite subsets of <span><math><mi>N</mi></math></span> containing 0 forms a monoid, with the singleton {0} as its neutral element.</div><div>We show that the only non-trivial automorphism of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mrow><mi>fin</mi></mrow><mo>,</mo><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> is the involution <span><math><mi>X</mi><mo>↦</mo><mi>max</mi><mo>⁡</mo><mi>X</mi><mo>−</mo><mi>X</mi></math></span>. The proof leverages ideas from additive number theory and proceeds through an unconventional induction on what we call the boxing dimension of a finite set of integers, that is, the smallest number of (discrete) intervals whose union is the set itself.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105961"},"PeriodicalIF":0.9,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On non-empty cross-t-intersecting families","authors":"Anshui Li ,&nbsp;Huajun Zhang","doi":"10.1016/j.jcta.2024.105960","DOIUrl":"10.1016/j.jcta.2024.105960","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be families of <em>k</em>-element subsets of a <em>n</em>-element set. We call them cross-<em>t</em>-intersecting if <span><math><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>|</mo><mo>≥</mo><mi>t</mi></math></span> for any <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> with <span><math><mi>i</mi><mo>≠</mo><mi>j</mi></math></span>. In this paper we will prove that, for <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi><mo>−</mo><mi>t</mi><mo>+</mo><mn>1</mn></math></span>, if <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> are non-empty cross-<em>t</em>-intersecting families, then<span><span><span><math><munder><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>m</mi></mrow></munder><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>≤</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><munder><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>t</mi><mo>−</mo><mn>1</mn></mrow></munder><mrow><mo>(</mo><mtable><mtr><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mi>i</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mi>i</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>m</mi><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>}</mo><mo>,</mo></math></span></span></span> where <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> is the size of the maximum <em>t</em>-intersecting family of <span><math><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></math></span>. Moreover, the extremal families attaining the upper bound are characterized.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105960"},"PeriodicalIF":0.9,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142316250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Avoiding intersections of given size in finite affine spaces AG(n,2)","authors":"Benedek Kovács ,&nbsp;Zoltán Lóránt Nagy","doi":"10.1016/j.jcta.2024.105959","DOIUrl":"10.1016/j.jcta.2024.105959","url":null,"abstract":"<div><div>We study the set of intersection sizes of a <em>k</em>-dimensional affine subspace and a point set of size <span><math><mi>m</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></math></span> of the <em>n</em>-dimensional binary affine space <span><math><mrow><mi>AG</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. Following the theme of Erdős, Füredi, Rothschild and T. Sós, we partially determine which local densities in <em>k</em>-dimensional affine subspaces are unavoidable in all <em>m</em>-element point sets in the <em>n</em>-dimensional affine space.</div><div>We also show constructions of point sets for which the intersection sizes with <em>k</em>-dimensional affine subspaces take values from a set of a small size compared to <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>. These are built up from affine subspaces and so-called subspace evasive sets. Meanwhile, we improve the best known upper bounds on subspace evasive sets and apply results concerning the canonical signed-digit (CSD) representation of numbers.</div><div><em>Keywords</em>: unavoidable, affine subspaces, evasive sets, random methods, canonical signed-digit number system.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105959"},"PeriodicalIF":0.9,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000980/pdfft?md5=62687b67d599290d3f204041642a9a6a&pid=1-s2.0-S0097316524000980-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A rank two Leonard pair in Terwilliger algebras of Doob graphs","authors":"John Vincent S. Morales","doi":"10.1016/j.jcta.2024.105958","DOIUrl":"10.1016/j.jcta.2024.105958","url":null,"abstract":"<div><div>Let <span><math><mi>Γ</mi><mo>=</mo><mi>Γ</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> denote the Doob graph formed by the Cartesian product of the <em>n</em>th Cartesian power of the Shrikhande graph and the <em>m</em>th Cartesian power of the complete graph on four vertices. Let <span><math><mi>T</mi><mo>=</mo><mi>T</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> denote the Terwilliger algebra of Γ with respect to a fixed vertex <em>x</em> of Γ and let <em>W</em> denote an arbitrary non-thin irreducible <em>T</em>-module in the standard module of Γ. In (Morales and Palma, 2021 <span><span>[25]</span></span>), it was shown that there exists a Lie algebra embedding <em>π</em> from the special orthogonal algebra <span><math><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> into <em>T</em> and that <em>W</em> is an irreducible <span><math><mi>π</mi><mo>(</mo><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo></math></span>-module. In this paper, we consider two Cartan subalgebras <span><math><mi>h</mi><mo>,</mo><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> of <span><math><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> such that <span><math><mi>h</mi><mo>,</mo><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> generate <span><math><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>. Using the embedding <span><math><mi>π</mi><mo>:</mo><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>→</mo><mi>T</mi></math></span>, we show that <span><math><mi>π</mi><mo>(</mo><mi>h</mi><mo>)</mo></math></span> and <span><math><mi>π</mi><mo>(</mo><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>)</mo></math></span> act on <em>W</em> as a rank two Leonard pair. We also obtain several direct sum decompositions of <em>W</em> akin to how split decompositions are obtained from Leonard pairs of rank one.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105958"},"PeriodicalIF":0.9,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Covering the set of p-elements in finite groups by proper subgroups","authors":"Attila Maróti ,&nbsp;Juan Martínez ,&nbsp;Alexander Moretó","doi":"10.1016/j.jcta.2024.105954","DOIUrl":"10.1016/j.jcta.2024.105954","url":null,"abstract":"<div><p>Let <em>p</em> be a prime and let <em>G</em> be a finite group which is generated by the set <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> of its <em>p</em>-elements. We show that if <em>G</em> is solvable and not a <em>p</em>-group, then the minimal number <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of proper subgroups of <em>G</em> whose union contains <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is equal to 1 less than the minimal number of proper subgroups of <em>G</em> whose union is <em>G</em>. For <em>p</em>-solvable groups <em>G</em>, we always have <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>p</mi><mo>+</mo><mn>1</mn></math></span>. We study the case of alternating and symmetric groups <em>G</em> in detail.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105954"},"PeriodicalIF":0.9,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000931/pdfft?md5=35b9a89a7b1644f6cad2cea930c20904&pid=1-s2.0-S0097316524000931-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Proofs of some conjectures of Merca on truncated series involving the Rogers-Ramanujan functions","authors":"Yongqiang Chen,&nbsp;Olivia X.M. Yao","doi":"10.1016/j.jcta.2024.105956","DOIUrl":"10.1016/j.jcta.2024.105956","url":null,"abstract":"<div><p>In 2012, Andrews and Merca investigated the truncated version of the Euler pentagonal number theorem. Their work has opened up a new study on truncated theta series and has inspired several mathematicians to work on the topic. In 2019, Merca studied the Rogers-Ramanujan functions and posed three groups of conjectures on truncated series involving the Rogers-Ramanujan functions. In this paper, we present a uniform method to prove the three groups of conjectures given by Merca based on a result due to Pólya and Szegö.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105956"},"PeriodicalIF":0.9,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142270376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the proportion of metric matroids whose Jacobians have nontrivial p-torsion","authors":"Sergio Ricardo Zapata Ceballos","doi":"10.1016/j.jcta.2024.105953","DOIUrl":"10.1016/j.jcta.2024.105953","url":null,"abstract":"<div><p>We study the proportion of metric matroids whose Jacobians have nontrivial <em>p</em>-torsion. We establish a correspondence between these Jacobians and the <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-rational points on configuration hypersurfaces, thereby relating their proportions. By counting points over finite fields, we prove that the proportion of these Jacobians is asymptotically equivalent to <span><math><mn>1</mn><mo>/</mo><mi>p</mi></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105953"},"PeriodicalIF":0.9,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S009731652400092X/pdfft?md5=d8e2893424d34795a1338a7aa80035a5&pid=1-s2.0-S009731652400092X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142243316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}