Journal of Combinatorial Theory Series A最新文献

{"title":"A multiparametric Murnaghan-Nakayama rule for Macdonald polynomials","authors":"Naihuan Jing ,&nbsp;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}

{"title":"A characterisation of edge-affine 2-arc-transitive covers of K2n,2n","authors":"Daniel R. Hawtin ,&nbsp;Cheryl E. Praeger ,&nbsp;Jin-Xin Zhou","doi":"10.1016/j.jcta.2024.105919","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105919","url":null,"abstract":"&lt;div&gt;&lt;p&gt;The family of finite 2-arc-transitive graphs of a given valency is closed under forming non-trivial &lt;em&gt;normal quotients&lt;/em&gt;, 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 &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. The characterisation identified the special role of graphs associated with a subgroup of automorphisms called an &lt;em&gt;n-dimensional mixed dihedral group&lt;/em&gt;. This is a group &lt;em&gt;H&lt;/em&gt; with two subgroups &lt;em&gt;X&lt;/em&gt; and &lt;em&gt;Y&lt;/em&gt;, each elementary abelian of order &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, such that &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;em&gt;H&lt;/em&gt; is generated by &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≅&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/p&gt;&lt;p&gt;Our characterisation shows that each 2-arc-transitive normal cover of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is either itself a Cayley graph, or is the line graph of a Cayley graph of an &lt;em&gt;n&lt;/em&gt;-dimensional mixed dihedral group. In the latter case, we show that the 2-arc-transitive group acting on the normal cover of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; induces an &lt;em&gt;edge-affine&lt;/em&gt; action on &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; (and we show that such actions are one of just four possible types of 2-arc-transitive actions on &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;). As a partial converse, we provide a graph theoretic characterisation of &lt;em&gt;n&lt;/em&gt;-dimensional mixed dihedral groups, and finally, for each &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, we give an explicit construction of an &lt;em&gt;n&lt;/em&gt;-dimensional mixed dihedral group which is a 2-group of order &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;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 ,&nbsp;Arseny Shur ,&nbsp;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>&gt;</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>&lt;</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>&lt;</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}

{"title":"Spectra of power hypergraphs and signed graphs via parity-closed walks","authors":"Lixiang Chen ,&nbsp;Edwin R. van Dam ,&nbsp;Changjiang Bu","doi":"10.1016/j.jcta.2024.105909","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105909","url":null,"abstract":"<div><p>The <em>k</em>-power hypergraph <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> is the <em>k</em>-uniform hypergraph that is obtained by adding <span><math><mi>k</mi><mo>−</mo><mn>2</mn></math></span> new vertices to each edge of a graph <em>G</em>, for <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>. A parity-closed walk in <em>G</em> is a closed walk that uses each edge an even number of times. In an earlier paper, we determined the eigenvalues of the adjacency tensor of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> using the eigenvalues of signed subgraphs of <em>G</em>. Here, we express the entire spectrum (that is, we determine all multiplicities and the characteristic polynomial) of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> in terms of parity-closed walks of <em>G</em>. Moreover, we give an explicit expression for the multiplicity of the spectral radius of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span>. As a side result, we show that the number of parity-closed walks of given length is the corresponding spectral moment averaged over all signed graphs with underlying graph <em>G</em>. By extrapolating the characteristic polynomial of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> to <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>, we introduce a pseudo-characteristic function which is shown to be the geometric mean of the characteristic polynomials of all signed graphs on <em>G</em>. This supplements a result by Godsil and Gutman that the arithmetic mean of the characteristic polynomials of all signed graphs on <em>G</em> equals the matching polynomial of <em>G</em>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"207 ","pages":"Article 105909"},"PeriodicalIF":1.1,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141078251","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":"Maximum flag-rank distance codes","authors":"Gianira N. Alfarano ,&nbsp;Alessandro Neri ,&nbsp;Ferdinando Zullo","doi":"10.1016/j.jcta.2024.105908","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105908","url":null,"abstract":"<div><p>In this paper we extend the study of linear spaces of upper triangular matrices endowed with the flag-rank metric. Such metric spaces are isometric to certain spaces of degenerate flags and have been suggested as suitable framework for network coding. In this setting we provide a Singleton-like bound which relates the parameters of a flag-rank-metric code. This allows us to introduce the family of maximum flag-rank distance codes, that are flag-rank-metric codes meeting the Singleton-like bound with equality. Finally, we provide several constructions of maximum flag-rank distance codes.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"207 ","pages":"Article 105908"},"PeriodicalIF":1.1,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000475/pdfft?md5=2c125aba1bc7cdaa56a76a0c7c4abe5d&pid=1-s2.0-S0097316524000475-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141078250","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":"Linkage of graphs with flows","authors":"Alex Abreu ,&nbsp;Marco Pacini ,&nbsp;Matheus Secco","doi":"10.1016/j.jcta.2024.105907","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105907","url":null,"abstract":"<div><p>We prove several linkage properties of graphs with flows, generalizing some results on linkage of graphs. This translates in properties of connectedness through codimension one of certain posets. For example, the poset of flows and the posets of odd and even tropical spin curves. These posets are, respectively, the posets underlying the moduli space of roots of divisors on tropical curves and the moduli spaces of odd and even tropical spin curves.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105907"},"PeriodicalIF":1.1,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140822507","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":"The enumeration of equivalent classes of minimal general dihedral group codes","authors":"Boheng Huang","doi":"10.1016/j.jcta.2024.105906","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105906","url":null,"abstract":"<div><p>A group code is a linear code which can be realized as a two-sided ideal of a group algebra over a finite field. When the characteristic of the field is prime to the order of the group, we will give explicit expressions for primitive central idempotents in the group algebra, which enables us to determine the number of equivalent classes of minimal group codes. Then, we apply our formula to calculate the number of equivalent classes of minimal general dihedral group codes.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105906"},"PeriodicalIF":1.1,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140649718","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":"The number of primitive words of unbounded exponent in the language of an HD0L-system is finite","authors":"Karel Klouda,&nbsp;Štěpán Starosta","doi":"10.1016/j.jcta.2024.105904","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105904","url":null,"abstract":"<div><p>Let <em>H</em> be an HD0L-system. We show that there are only finitely many primitive words <em>v</em> with the property that <span><math><msup><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>, for all integers <em>k</em>, is an element of the factorial language of <em>H</em>. In particular, this result applies to the set of all factors of a morphic word. We provide a formalized proof in the proof assistant Isabelle/HOL as part of the Combinatorics on Words Formalized project.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105904"},"PeriodicalIF":1.1,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140633182","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":"Basic tetravalent oriented graphs with cyclic normal quotients","authors":"Nemanja Poznanović ,&nbsp;Cheryl E. Praeger","doi":"10.1016/j.jcta.2024.105895","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105895","url":null,"abstract":"<div><p>Let <span><math><mi>OG</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span> denote the family of all graph-group pairs <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><mi>G</mi><mo>)</mo></math></span> where Γ is finite, 4-valent, connected, and <em>G</em>-oriented (<em>G</em>-half-arc-transitive). A subfamily of <span><math><mi>OG</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span> has recently been identified as ‘basic’ in the sense that all graphs in this family are normal covers of at least one basic member. In this paper we provide a description of such basic pairs which have at least one <em>G</em>-normal quotient which is isomorphic to a cycle graph. In doing so, we produce many new infinite families of examples and solve several problems posed in the recent literature on this topic. This result completes a research project aiming to provide a description of all basic pairs in <span><math><mi>OG</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105895"},"PeriodicalIF":1.1,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000347/pdfft?md5=f4ddc34a4bd5053c5b3e0115d6f19233&pid=1-s2.0-S0097316524000347-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140558761","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":"Explicit formulas for a family of hypermaps beyond the one-face case","authors":"Zi-Wei Bai,&nbsp;Ricky X.F. Chen","doi":"10.1016/j.jcta.2024.105905","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105905","url":null,"abstract":"<div><p>Enumeration of hypermaps (or Grothendieck's dessins d'enfants) is widely studied in many fields. In particular, enumerating hypermaps with a fixed edge-type according to the number of faces and genus is one topic of great interest. The first systematic study of hypermaps with one face and any fixed edge-type is the work of Jackson (1987) <span>[23]</span> using group characters. Stanley later (2011) obtained the genus distribution polynomial of one-face hypermaps of any fixed edge-type expressed in terms of the backward shift operator. There is also enormous amount of work on enumerating one-face hypermaps of specific edge-types. Hypermaps with more faces are generally much harder to enumerate and results are rare. Our main results here are formulas for the genus distribution polynomials for a family of typical two-face hypermaps including almost all edge-types, the purely imaginary zeros property of these polynomials, and the log-concavity of the coefficients.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105905"},"PeriodicalIF":1.1,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140604786","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}