{"title":"Cluster braid groups of Coxeter-Dynkin diagrams","authors":"Zhe Han , Ping He , Yu Qiu","doi":"10.1016/j.jcta.2024.105935","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105935","url":null,"abstract":"<div><p>Cluster exchange groupoids are introduced by King-Qiu as an enhancement of cluster exchange graphs to study stability conditions and quadratic differentials. In this paper, we introduce the cluster exchange groupoid for any finite Coxeter-Dynkin diagram Δ and show that its fundamental group is isomorphic to the corresponding braid group associated with Δ.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"208 ","pages":"Article 105935"},"PeriodicalIF":0.9,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141582036","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}
Rigoberto Flórez , José L. Ramírez , Diego Villamizar
{"title":"Restricted bargraphs and unimodal compositions","authors":"Rigoberto Flórez , José L. Ramírez , Diego Villamizar","doi":"10.1016/j.jcta.2024.105934","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105934","url":null,"abstract":"<div><p>In this paper, we present a study on <em>polyominoes</em>, which are polygons created by connecting unit squares along their edges. Specifically, we focus on a related concept called a <em>bargraph</em>, which is a path on a lattice in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo>×</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></math></span> traced along the boundaries of a column convex polyomino where the lower edge is on the <em>x</em>-axis. To explore new variations of bargraphs, we introduce the notion of <em>non-decreasing bargraphs</em>, which incorporate an additional restriction concerning the valleys within the path. We establish intriguing connections between these novel objects and unimodal compositions. To facilitate our analysis, we employ generating functions, including <em>q</em>-series, as well as various closed formulas. These tools enable us to enumerate the different types of bargraphs based on their semi-perimeter, area, and the number of peaks. Furthermore, we provide combinatorial justifications for some of the derived closed formulas.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"208 ","pages":"Article 105934"},"PeriodicalIF":0.9,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000736/pdfft?md5=f5366b9dc5560c0148e0644514e1990d&pid=1-s2.0-S0097316524000736-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141541064","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":"Positivity and tails of pentagonal number series","authors":"Nian Hong Zhou","doi":"10.1016/j.jcta.2024.105933","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105933","url":null,"abstract":"<div><p>In this paper, we refine a result of Andrews and Merca on truncated pentagonal number series. Subsequently, we establish some positivity results involving Andrews–Gordon–Bressoud identities and <em>d</em>-regular partitions. In particular, we prove several conjectures of Merca and Krattenthaler–Merca–Radu on truncated pentagonal number series.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"208 ","pages":"Article 105933"},"PeriodicalIF":0.9,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141541065","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}
Sucharita Biswas , Peter J. Cameron , Angsuman Das , Hiranya Kishore Dey
{"title":"On the difference of the enhanced power graph and the power graph of a finite group","authors":"Sucharita Biswas , Peter J. Cameron , Angsuman Das , Hiranya Kishore Dey","doi":"10.1016/j.jcta.2024.105932","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105932","url":null,"abstract":"<div><p>The difference graph <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a finite group <em>G</em> is the difference of the enhanced power graph of <em>G</em> and the power graph of <em>G</em>, where all isolated vertices are removed. In this paper we study the connectedness and perfectness of <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> with respect to various properties of the underlying group <em>G</em>. We also find several connections between the difference graph of <em>G</em> and the Gruenberg-Kegel graph of <em>G</em>. We also examine the operation of twin reduction on graphs, a technique which produces smaller graphs which may be easier to analyze. Applying this technique to simple groups can have a number of outcomes, not fully understood, but including some graphs with large girth.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"208 ","pages":"Article 105932"},"PeriodicalIF":0.9,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141439067","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":"Flag-transitive automorphism groups of 2-designs with λ ≥ (r,λ)2 are not product type","authors":"Huiling Li , Zhilin Zhang , Shenglin Zhou","doi":"10.1016/j.jcta.2024.105923","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105923","url":null,"abstract":"<div><p>In this note we show that a flag-transitive automorphism group <em>G</em> of a non-trivial 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> design with <span><math><mi>λ</mi><mo>≥</mo><msup><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>λ</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> is not of product action type. In conclusion, <em>G</em> is a primitive group of affine or almost simple type.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"208 ","pages":"Article 105923"},"PeriodicalIF":1.1,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141429448","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":"Intersection density of imprimitive groups of degree pq","authors":"Angelot Behajaina , Roghayeh Maleki , Andriaherimanana Sarobidy Razafimahatratra","doi":"10.1016/j.jcta.2024.105922","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105922","url":null,"abstract":"<div><p>A subset <span><math><mi>F</mi></math></span> of a finite transitive group <span><math><mi>G</mi><mo>≤</mo><mi>Sym</mi><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> is <em>intersecting</em> if any two elements of <span><math><mi>F</mi></math></span> agree on an element of Ω. The <em>intersection density</em> of <em>G</em> is the number<span><span><span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo></mo><mrow><mo>{</mo><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow><mo>/</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mo>|</mo><mo>|</mo><mi>F</mi><mo>⊂</mo><mi>G</mi><mtext> is intersecting</mtext><mo>}</mo></mrow><mo>,</mo></math></span></span></span> where <span><math><mi>ω</mi><mo>∈</mo><mi>Ω</mi></math></span> and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span> is the stabilizer of <em>ω</em> in <em>G</em>. It is known that if <span><math><mi>G</mi><mo>≤</mo><mi>Sym</mi><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> is an imprimitive group of degree a product of two odd primes <span><math><mi>p</mi><mo>></mo><mi>q</mi></math></span> admitting a block of size <em>p</em> or two complete block systems, whose blocks are of size <em>q</em>, then <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>.</p><p>In this paper, we analyze the intersection density of imprimitive groups of degree <em>pq</em> with a unique block system with blocks of size <em>q</em> based on the kernel of the induced action on blocks. For those whose kernels are non-trivial, it is proved that the intersection density is larger than 1 whenever there exists a cyclic code <em>C</em> with parameters <span><math><msub><mrow><mo>[</mo><mi>p</mi><mo>,</mo><mi>k</mi><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> such that any codeword of <em>C</em> has weight at most <span><math><mi>p</mi><mo>−</mo><mn>1</mn></math></span>, and under some additional conditions on the cyclic code, it is a proper rational number. For those that are quasiprimitive, we reduce the cases to almost simple groups containing <span><math><mi>Alt</mi><mo>(</mo><mn>5</mn><mo>)</mo></math></span> or a projective special linear group. We give some examples where the latter has intersection density equal to 1, under some restrictions on <em>p</em> and <em>q</em>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"208 ","pages":"Article 105922"},"PeriodicalIF":1.1,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S009731652400061X/pdfft?md5=4600ca58b59525e76de9f361f9c870b7&pid=1-s2.0-S009731652400061X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141308391","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":"Remarks on MacMahon's q-series","authors":"Ken Ono, Ajit Singh","doi":"10.1016/j.jcta.2024.105921","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105921","url":null,"abstract":"<div><p>In his important 1920 paper on partitions, MacMahon defined the partition generating functions<span><span><span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></munderover><mi>m</mi><mo>(</mo><mi>k</mi><mo>;</mo><mi>n</mi><mo>)</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>:</mo><mo>=</mo><munder><mo>∑</mo><mrow><mn>0</mn><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></munder><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup></mrow><mrow><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>⋯</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>,</mo></math></span></span></span><span><span><span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></munderover><msub><mrow><mi>m</mi></mrow><mrow><mi>odd</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>;</mo><mi>n</mi><mo>)</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>:</mo><mo>=</mo><munder><mo>∑</mo><mrow><mn>0</mn><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></munder><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>2</mn><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><mn>2</mn><msub><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>−</mo><mi>k</mi></mrow></msup></mrow><mrow><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mn>1<","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"207 ","pages":"Article 105921"},"PeriodicalIF":1.1,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141240350","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":"A multiparametric Murnaghan-Nakayama rule for Macdonald polynomials","authors":"Naihuan Jing , Ning Liu","doi":"10.1016/j.jcta.2024.105920","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105920","url":null,"abstract":"<div><p>We introduce a new family of operators as multi-parameter deformation of the one-row Macdonald polynomials. The matrix coefficients of these operators acting on the space of symmetric functions with rational coefficients in two parameters <span><math><mi>q</mi><mo>,</mo><mi>t</mi></math></span> (denoted by <span><math><mi>Λ</mi><mo>(</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>) are computed by assigning some values to skew Macdonald polynomials in <em>λ</em>-ring notation. The new rule is utilized to provide new iterative formulas and also recover various existing formulas in a unified manner. Specifically the following applications are discussed: (i) A <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>-Murnaghan-Nakayama rule for Macdonald functions is given as a generalization of the <em>q</em>-Murnaghan-Nakayama rule; (ii) An iterative formula for the <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>-Green polynomial is deduced; (iii) A simple proof of the Murnaghan-Nakayama rule for the Hecke algebra and the Hecke-Clifford algebra is offered; (iv) A combinatorial inversion of the Pieri rule for Hall-Littlewood functions is derived with the help of the vertex operator realization of the Hall-Littlewood functions; (v) Two iterative formulae for the <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>-Kostka polynomials <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>λ</mi><mi>μ</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> are obtained from the dual version of our multiparametric Murnaghan-Nakayama rule, one of which yields an explicit formula for arbitrary <em>λ</em> and <em>μ</em> in terms of the generalized <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>-binomial coefficient introduced independently by Lassalle and Okounkov.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"207 ","pages":"Article 105920"},"PeriodicalIF":1.1,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141163388","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}
Daniel R. Hawtin , Cheryl E. Praeger , Jin-Xin Zhou
{"title":"A characterisation of edge-affine 2-arc-transitive covers of K2n,2n","authors":"Daniel R. Hawtin , Cheryl E. Praeger , Jin-Xin Zhou","doi":"10.1016/j.jcta.2024.105919","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105919","url":null,"abstract":"<div><p>The family of finite 2-arc-transitive graphs of a given valency is closed under forming non-trivial <em>normal quotients</em>, and graphs in this family having no non-trivial normal quotient are called ‘basic’. To date, the vast majority of work in the literature has focused on classifying these ‘basic’ graphs. By contrast we give here a characterisation of the normal covers of the ‘basic’ 2-arc-transitive graphs <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>. The characterisation identified the special role of graphs associated with a subgroup of automorphisms called an <em>n-dimensional mixed dihedral group</em>. This is a group <em>H</em> with two subgroups <em>X</em> and <em>Y</em>, each elementary abelian of order <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>, such that <span><math><mi>X</mi><mo>∩</mo><mi>Y</mi><mo>=</mo><mn>1</mn></math></span>, <em>H</em> is generated by <span><math><mi>X</mi><mo>∪</mo><mi>Y</mi></math></span>, and <span><math><mi>H</mi><mo>/</mo><msup><mrow><mi>H</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≅</mo><mi>X</mi><mo>×</mo><mi>Y</mi></math></span>.</p><p>Our characterisation shows that each 2-arc-transitive normal cover of <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> is either itself a Cayley graph, or is the line graph of a Cayley graph of an <em>n</em>-dimensional mixed dihedral group. In the latter case, we show that the 2-arc-transitive group acting on the normal cover of <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> induces an <em>edge-affine</em> action on <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> (and we show that such actions are one of just four possible types of 2-arc-transitive actions on <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>). As a partial converse, we provide a graph theoretic characterisation of <em>n</em>-dimensional mixed dihedral groups, and finally, for each <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, we give an explicit construction of an <em>n</em>-dimensional mixed dihedral group which is a 2-group of order <span><math><msup><mrow><mn>2</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><m","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"207 ","pages":"Article 105919"},"PeriodicalIF":1.1,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141163387","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":"Power-free complementary binary morphisms","authors":"Jeffrey Shallit , Arseny Shur , Stefan Zorcic","doi":"10.1016/j.jcta.2024.105910","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105910","url":null,"abstract":"<div><p>We revisit the topic of power-free morphisms, focusing on the properties of the class of complementary morphisms. Such morphisms are defined over a 2-letter alphabet, and map the letters 0 and 1 to complementary words. We prove that every prefix of the famous Thue–Morse word <strong>t</strong> gives a complementary morphism that is <span><math><msup><mrow><mn>3</mn></mrow><mrow><mo>+</mo></mrow></msup></math></span>-free and hence <em>α</em>-free for every real number <span><math><mi>α</mi><mo>></mo><mn>3</mn></math></span>. We also describe, using a 4-state binary finite automaton, the lengths of all prefixes of <strong>t</strong> that give cubefree complementary morphisms. Next, we show that 3-free (cubefree) complementary morphisms of length <em>k</em> exist for all <span><math><mi>k</mi><mo>∉</mo><mo>{</mo><mn>3</mn><mo>,</mo><mn>6</mn><mo>}</mo></math></span>. Moreover, if <em>k</em> is not of the form <span><math><mn>3</mn><mo>⋅</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>, then the images of letters can be chosen to be factors of <strong>t</strong>. Finally, we observe that each cubefree complementary morphism is also <em>α</em>-free for some <span><math><mi>α</mi><mo><</mo><mn>3</mn></math></span>; in contrast, no binary morphism that maps each letter to a word of length 3 (resp., a word of length 6) is <em>α</em>-free for any <span><math><mi>α</mi><mo><</mo><mn>3</mn></math></span>.</p><p>In addition to more traditional techniques of combinatorics on words, we also rely on the Walnut theorem-prover. Its use and limitations are discussed.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"207 ","pages":"Article 105910"},"PeriodicalIF":1.1,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141083216","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}
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