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The Merino–Welsh Conjecture for Split Matroids 分裂拟阵的Merino–Welsh猜想
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-12-17 DOI: 10.1007/s00026-022-00628-w
Luis Ferroni, Benjamin Schröter
{"title":"The Merino–Welsh Conjecture for Split Matroids","authors":"Luis Ferroni,&nbsp;Benjamin Schröter","doi":"10.1007/s00026-022-00628-w","DOIUrl":"10.1007/s00026-022-00628-w","url":null,"abstract":"<div><p>In 1999, Merino and Welsh conjectured that evaluations of the Tutte polynomial of a graph satisfy an inequality. In this short article, we show that the conjecture generalized to matroids holds for the large class of all split matroids by exploiting the structure of their lattice of cyclic flats. This class of matroids strictly contains all paving and copaving matroids.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-022-00628-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46472179","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: 3
A Combinatorial Proof of the Unimodality and Symmetry of Weak Composition Rank Sequences 弱组合秩序列的单模态与对称性的组合证明
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-12-11 DOI: 10.1007/s00026-022-00624-0
Yueming Zhong
{"title":"A Combinatorial Proof of the Unimodality and Symmetry of Weak Composition Rank Sequences","authors":"Yueming Zhong","doi":"10.1007/s00026-022-00624-0","DOIUrl":"10.1007/s00026-022-00624-0","url":null,"abstract":"<div><p>A weak composition of an integer <i>s</i> with <i>m</i> parts is a way of writing <i>s</i> as the sum of a sequence of non-negative integers of length <i>m</i>. Given two positive integers <i>m</i> and <i>n</i>, let <i>N</i>(<i>m</i>, <i>n</i>) denote the set of all weak compositions <span>(alpha =(alpha _1,dots ,alpha _m))</span> with <span>(0 le alpha _i le n)</span> for <span>(1 le i le m)</span> and <span>(c_w^{m,n}(s))</span> be the number of weak composition of <i>s</i> into <i>m</i> parts with no part exceeding <i>n</i>. A poset is called a symmetric chain decomposition if the poset can be expressed as a disjoint union of symmetric chains. In this paper, we show that the poset <i>N</i>(<i>m</i>, <i>n</i>) can be expressed as a disjoint of symmetric chains by constructive method, which implies that its rank sequence <span>(c_w^{m,n}(0),c_w^{m,n}(1),dots ,c_w^{m,n}(mn))</span> is unimodal and symmetric.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44914671","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
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":null,"pages":null},"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
Cluster Scattering Diagrams and Theta Functions for Reciprocal Generalized Cluster Algebras 互易广义簇代数的簇散射图和Theta函数
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-12-09 DOI: 10.1007/s00026-022-00623-1
Man-Wai Cheung, Elizabeth Kelley, Gregg Musiker
{"title":"Cluster Scattering Diagrams and Theta Functions for Reciprocal Generalized Cluster Algebras","authors":"Man-Wai Cheung,&nbsp;Elizabeth Kelley,&nbsp;Gregg Musiker","doi":"10.1007/s00026-022-00623-1","DOIUrl":"10.1007/s00026-022-00623-1","url":null,"abstract":"<div><p>We give a construction of generalized cluster varieties and generalized cluster scattering diagrams for reciprocal generalized cluster algebras, the latter of which were defined by Chekhov and Shapiro. These constructions are analogous to the structures given for ordinary cluster algebras in the work of Gross, Hacking, Keel, and Kontsevich. As a consequence of these constructions, we are also able to construct theta functions for generalized cluster algebras, again in the reciprocal case, and demonstrate a number of their structural properties.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43533366","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: 5
The Minimal Sum of Squares Over Partitions with a Nonnegative Rank 非负秩分区上的最小平方和
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-12-03 DOI: 10.1007/s00026-022-00625-z
Sela Fried
{"title":"The Minimal Sum of Squares Over Partitions with a Nonnegative Rank","authors":"Sela Fried","doi":"10.1007/s00026-022-00625-z","DOIUrl":"10.1007/s00026-022-00625-z","url":null,"abstract":"<div><p>Motivated by a question of Defant and Propp (Electron J Combin 27:Article P3.51, 2020) regarding the connection between the degrees of noninvertibility of functions and those of their iterates, we address the combinatorial optimization problem of minimizing the sum of squares over partitions of <i>n</i> with a nonnegative rank. Denoting the sequence of the minima by <span>((m_n)_{nin {mathbb {N}}})</span>, we prove that <span>(m_n=Theta left( n^{4/3}right) )</span>. Consequently, we improve by a factor of 2 the lower bound provided by Defant and Propp for iterates of order two.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50444826","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
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":null,"pages":null},"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
Large (p)-Core (p')-Partitions and Walks on the Additive Residue Graph 大型$$p$$ -Core $$p'$$ -加性残差图上的分区和行走
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-11-24 DOI: 10.1007/s00026-022-00622-2
Eoghan McDowell
{"title":"Large (p)-Core (p')-Partitions and Walks on the Additive Residue Graph","authors":"Eoghan McDowell","doi":"10.1007/s00026-022-00622-2","DOIUrl":"10.1007/s00026-022-00622-2","url":null,"abstract":"<div><p>This paper investigates partitions which have neither parts nor hook lengths divisible by <span>(p)</span>, referred to as <span>(p)</span>-core <span>(p')</span>-partitions. We show that the largest <span>(p)</span>-core <span>(p')</span>-partition corresponds to the longest walk on a graph with vertices <span>({0, 1, ldots , p-1})</span> and labelled edges defined via addition modulo <span>(p)</span>. We also exhibit an explicit family of large <span>(p)</span>-core <span>(p')</span>-partitions, giving a lower bound on the size of the largest such partition which is of the same degree as the upper bound found by McSpirit and Ono.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45006429","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
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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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
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