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Boolean Complexes of Involutions 对合的布尔复形
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-12-28 DOI: 10.1007/s00026-022-00629-9
Axel Hultman, Vincent Umutabazi
{"title":"Boolean Complexes of Involutions","authors":"Axel Hultman,&nbsp;Vincent Umutabazi","doi":"10.1007/s00026-022-00629-9","DOIUrl":"10.1007/s00026-022-00629-9","url":null,"abstract":"<div><p>Let (<i>W</i>, <i>S</i>) be a Coxeter system. We introduce the boolean complex of involutions of <i>W</i> which is an analogue of the boolean complex of <i>W</i> studied by Ragnarsson and Tenner. By applying discrete Morse theory, we determine the homotopy type of the boolean complex of involutions for a large class of (<i>W</i>, <i>S</i>), including all finite Coxeter groups, finding that the homotopy type is that of a wedge of spheres of dimension <span>(vert Svert -1)</span>. In addition, we find simple recurrence formulas for the number of spheres in the wedge.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 1","pages":"129 - 147"},"PeriodicalIF":0.5,"publicationDate":"2022-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-022-00629-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41326138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Equivariant Euler Characteristics of Subgroup Complexes of Symmetric Groups 对称群子群配合物的等变Euler特性
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-12-17 DOI: 10.1007/s00026-022-00630-2
Zhipeng Duan
{"title":"Equivariant Euler Characteristics of Subgroup Complexes of Symmetric Groups","authors":"Zhipeng Duan","doi":"10.1007/s00026-022-00630-2","DOIUrl":"10.1007/s00026-022-00630-2","url":null,"abstract":"<div><p>Equivariant Euler characteristics are important numerical homotopy invariants for objects with group actions. They have deep connections with many other areas like modular representation theory and chromatic homotopy theory. They are also computable, especially for combinatorial objects like partition posets, buildings associated with finite groups of Lie types, etc. In this article, we make new contributions to concrete computations by determining the equivariant Euler characteristics for all subgroup complexes of symmetric groups <span>(varSigma _n)</span> when n is prime, twice a prime, or a power of two and several variants. There are two basic approaches to calculating equivariant Euler characteristics. One is based on a recursion formula and generating functions, and another on analyzing the fixed points of abelian subgroups. In this article, we adopt the second approach since the fixed points of abelian subgroups are simple in this case.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 1","pages":"67 - 85"},"PeriodicalIF":0.5,"publicationDate":"2022-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48647046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Defining Binary Phylogenetic Trees Using Parsimony 用简约法定义二元系统发育树
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-12-17 DOI: 10.1007/s00026-022-00627-x
Mareike Fischer
{"title":"Defining Binary Phylogenetic Trees Using Parsimony","authors":"Mareike Fischer","doi":"10.1007/s00026-022-00627-x","DOIUrl":"10.1007/s00026-022-00627-x","url":null,"abstract":"<div><p>Phylogenetic (i.e., leaf-labeled) trees play a fundamental role in evolutionary research. A typical problem is to reconstruct such trees from data like DNA alignments (whose columns are often referred to as characters), and a simple optimization criterion for such reconstructions is maximum parsimony. It is generally assumed that this criterion works well for data in which state changes are rare. In the present manuscript, we prove that each binary phylogenetic tree <i>T</i> with <span>(nge 20 k)</span> leaves is uniquely defined by the set <span>(A_k(T))</span>, which consists of all characters with parsimony score <i>k</i> on <i>T</i>. This can be considered as a promising first step toward showing that maximum parsimony as a tree reconstruction criterion is justified when the number of changes in the data is relatively small.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 3","pages":"457 - 467"},"PeriodicalIF":0.5,"publicationDate":"2022-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-022-00627-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43287985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On the Homeomorphism and Homotopy Type of Complexes of Multichains 关于多链配合物的同胚性和同伦型
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-12-11 DOI: 10.1007/s00026-022-00626-y
Shaheen Nazir, Volkmar Welker
{"title":"On the Homeomorphism and Homotopy Type of Complexes of Multichains","authors":"Shaheen Nazir,&nbsp;Volkmar Welker","doi":"10.1007/s00026-022-00626-y","DOIUrl":"10.1007/s00026-022-00626-y","url":null,"abstract":"<div><p>In this paper we define and study for a finite partially ordered set <i>P</i> a class of simplicial complexes on the set <span>(P_r)</span> of <i>r</i>-element multichains of <i>P</i>. The simplicial complexes depend on a strictly monotone function from [<i>r</i>] to [2<i>r</i>]. We show that there are exactly <span>(2^r)</span> such functions which yield subdivisions of the order complex of <i>P</i>, of which <span>(2^{r-1})</span> are pairwise different. Within this class are, for example, the order complexes of the intervals in <i>P</i>, the zig-zag poset of <i>P</i>, and the <span>(r{hbox {th}})</span> edgewise subdivision of the order complex of <i>P</i>. We also exhibit a large subclass for which our simplicial complexes are order complexes and homotopy equivalent to the order complex of <i>P</i>.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 2","pages":"229 - 247"},"PeriodicalIF":0.5,"publicationDate":"2022-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46322893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Growing Random Uniform d-ary Trees 生长随机均匀树
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-11-29 DOI: 10.1007/s00026-022-00621-3
Jean-François Marckert
{"title":"Growing Random Uniform d-ary Trees","authors":"Jean-François Marckert","doi":"10.1007/s00026-022-00621-3","DOIUrl":"10.1007/s00026-022-00621-3","url":null,"abstract":"<div><p>Let <span>({{mathcal {T}}}_{d}(n))</span> be the set of <i>d</i>-ary rooted trees with <i>n</i> internal nodes. We give a method to construct a sequence <span>(( textbf{t}_{n},nge 0))</span>, where, for any <span>(nge 1)</span>, <span>( textbf{t}_{n})</span> has the uniform distribution in <span>({{mathcal {T}}}_{d}(n))</span>, and <span>( textbf{t}_{n})</span> is constructed from <span>( textbf{t}_{n-1})</span> by the addition of a new node, and a rearrangement of the structure of <span>( textbf{t}_{n-1})</span>. This method is inspired by Rémy’s algorithm which does this job in the binary case, but it is different from it. This provides a method for the random generation of a uniform <i>d</i>-ary tree in <span>({{mathcal {T}}}_{d}(n))</span> with a cost linear in <i>n</i>.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 1","pages":"51 - 66"},"PeriodicalIF":0.5,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45798870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
The Limit Theorem with Respect to the Matrices on Non-backtracking Paths of a Graph 图的非回溯路径上矩阵的极限定理
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-11-15 DOI: 10.1007/s00026-022-00617-z
Takehiro Hasegawa, Takashi Komatsu, Norio Konno, Hayato Saigo, Seiken Saito, Iwao Sato, Shingo Sugiyama
{"title":"The Limit Theorem with Respect to the Matrices on Non-backtracking Paths of a Graph","authors":"Takehiro Hasegawa,&nbsp;Takashi Komatsu,&nbsp;Norio Konno,&nbsp;Hayato Saigo,&nbsp;Seiken Saito,&nbsp;Iwao Sato,&nbsp;Shingo Sugiyama","doi":"10.1007/s00026-022-00617-z","DOIUrl":"10.1007/s00026-022-00617-z","url":null,"abstract":"<div><p>We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the <i>k</i>th moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the <span>(p^m)</span>th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 2","pages":"249 - 268"},"PeriodicalIF":0.5,"publicationDate":"2022-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45107650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Upper Bounds on the Smallest Positive Eigenvalue of Trees 树的最小正特征值的上界
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-11-09 DOI: 10.1007/s00026-022-00619-x
Sonu Rani, Sasmita Barik
{"title":"Upper Bounds on the Smallest Positive Eigenvalue of Trees","authors":"Sonu Rani,&nbsp;Sasmita Barik","doi":"10.1007/s00026-022-00619-x","DOIUrl":"10.1007/s00026-022-00619-x","url":null,"abstract":"<div><p>In this article, we undertake the problem of finding the first four trees on a fixed number of vertices with the maximum smallest positive eigenvalue. Let <span>({mathcal {T}}_{n,d})</span> denote the class of trees on <i>n</i> vertices with diameter <i>d</i>. First, we obtain the bounds on the smallest positive eigenvalue of trees in <span>({mathcal {T}}_{n,d})</span> for <span>(d =2,3,4)</span> and then upper bounds on the smallest positive eigenvalue of trees are obtained in general class of all trees on <i>n</i> vertices. Finally, the first four trees on <i>n</i> vertices with the maximum, second maximum, third maximum and fourth maximum smallest positive eigenvalue are characterized.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 1","pages":"19 - 29"},"PeriodicalIF":0.5,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-022-00619-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46906319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On Two-Sided Cayley Graphs of Semigroups and Groups 半群和群的双面Cayley图
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-11-06 DOI: 10.1007/s00026-022-00618-y
Farshad Hassani Hajivand, Behnam Khosravi
{"title":"On Two-Sided Cayley Graphs of Semigroups and Groups","authors":"Farshad Hassani Hajivand,&nbsp;Behnam Khosravi","doi":"10.1007/s00026-022-00618-y","DOIUrl":"10.1007/s00026-022-00618-y","url":null,"abstract":"<div><p>In this paper, first we introduce the notion of two-sided Cayley graph of a semigroup. Then, we investigate some fundamental properties of these graphs and we use our results to give partial answers to some problems raised by Iradmusa and Praeger about two-sided group graphs (two-sided Cayley graphs of groups). Specially, as a consequence of our results, we determine all undirected two-sided Cayley graphs of groups which are connected. Furthermore, by introducing the notion of color-preserving automorphisms of a two-sided Cayley graph of a semigroup (group) and calculating them under some assumptions, we determine the family of color-vertex transitive two-sided Cayley graphs of semigroups (groups).\u0000</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 2","pages":"413 - 432"},"PeriodicalIF":0.5,"publicationDate":"2022-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41309118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Sandpile Groups of Random Bipartite Graphs 随机二部图的沙堆群
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-10-29 DOI: 10.1007/s00026-022-00616-0
Shaked Koplewitz
{"title":"Sandpile Groups of Random Bipartite Graphs","authors":"Shaked Koplewitz","doi":"10.1007/s00026-022-00616-0","DOIUrl":"10.1007/s00026-022-00616-0","url":null,"abstract":"<div><p>We determine the asymptotic distribution of the <i>p</i>-rank of the sandpile groups of random bipartite graphs. We see that this depends on the ratio between the number of vertices on each side, with a threshold when the ratio between the sides is equal to <span>(frac{1}{p})</span>. We follow the approach of Wood (J Am Math Soc 30(4):915–958, 2017) and consider random graphs as a special case of random matrices, and rely on a variant the definition of min-entropy given by Maples (Cokernels of random matrices satisfy the Cohen–Lenstra heuristics, 2013) to obtain useful results about these random matrices. Our results show that unlike the sandpile groups of Erdős–Rényi random graphs, the distribution of the sandpile groups of random bipartite graphs depends on the properties of the graph, rather than coming from some more general random group model.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 1","pages":"1 - 18"},"PeriodicalIF":0.5,"publicationDate":"2022-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47887705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Asymptotics, Turán Inequalities, and the Distribution of the BG-Rank and 2-Quotient Rank of Partitions 渐近性，Turán不等式，以及分区的bg秩和2商秩的分布
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-10-29 DOI: 10.1007/s00026-022-00612-4
Andrew Baker, Joshua Males
{"title":"Asymptotics, Turán Inequalities, and the Distribution of the BG-Rank and 2-Quotient Rank of Partitions","authors":"Andrew Baker,&nbsp;Joshua Males","doi":"10.1007/s00026-022-00612-4","DOIUrl":"10.1007/s00026-022-00612-4","url":null,"abstract":"<div><p>Let <i>j</i>, <i>n</i> be even positive integers, and let <span>(overline{p}_j(n))</span> denote the number of partitions with BG-rank <i>j</i>, and <span>(overline{p}_j(a,b;n))</span> to be the number of partitions with BG-rank <i>j</i> and 2-quotient rank congruent to <span>(a , left( mathrm {mod} , b right) )</span>. We give asymptotics for both statistics, and show that <span>(overline{p}_j(a,b;n))</span> is asymptotically equidistributed over the congruence classes modulo <i>b</i>. We also show that each of <span>(overline{p}_j(n))</span> and <span>(overline{p}_j(a,b;n))</span> asymptotically satisfy all higher-order Turán inequalities.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 4","pages":"769 - 780"},"PeriodicalIF":0.5,"publicationDate":"2022-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48590335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
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