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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
On Perfect Sequence Covering Arrays 关于完全序列覆盖数组
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-11-01 DOI: 10.1007/s00026-022-00610-6
Aidan R. Gentle, Ian M. Wanless
{"title":"On Perfect Sequence Covering Arrays","authors":"Aidan R. Gentle,&nbsp;Ian M. Wanless","doi":"10.1007/s00026-022-00610-6","DOIUrl":"10.1007/s00026-022-00610-6","url":null,"abstract":"<div><p>A PSCA<span>((v, t, lambda ))</span> is a multiset of permutations of the <i>v</i>-element alphabet <span>({0, dots , v-1})</span>, such that every sequence of <i>t</i> distinct elements of the alphabet appears in the specified order in exactly <span>(lambda )</span> of the permutations. For <span>(v geqslant t geqslant 2)</span>, we define <i>g</i>(<i>v</i>, <i>t</i>) to be the smallest positive integer <span>(lambda )</span>, such that a PSCA<span>((v, t, lambda ))</span> exists. We show that <span>(g(6, 3) = g(7, 3) = g(7, 4) = 2)</span> and <span>(g(8, 3) = 3)</span>. Using suitable permutation representations of groups, we make improvements to the upper bounds on <i>g</i>(<i>v</i>, <i>t</i>) for many values of <span>(v leqslant 32)</span> and <span>(3leqslant tleqslant 6)</span>. We also prove a number of restrictions on the distribution of symbols among the columns of a PSCA.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-022-00610-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43061045","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: 2
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":null,"pages":null},"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":null,"pages":null},"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
Ramanujan’s Theta Functions and Parity of Parts and Cranks of Partitions Ramanujan的Theta函数与分部的奇偶性和分区的曲柄。
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-10-25 DOI: 10.1007/s00026-022-00615-1
Koustav Banerjee, Manosij Ghosh Dastidar
{"title":"Ramanujan’s Theta Functions and Parity of Parts and Cranks of Partitions","authors":"Koustav Banerjee,&nbsp;Manosij Ghosh Dastidar","doi":"10.1007/s00026-022-00615-1","DOIUrl":"10.1007/s00026-022-00615-1","url":null,"abstract":"<div><p>In this paper, we explore intricate connections between Ramanujan’s theta functions and a class of partition functions defined by the nature of the parity of their parts. This consequently leads us to the parity analysis of the crank of a partition and its correlation with the number of partitions with odd number of parts, self-conjugate partitions, and also with Durfee squares and Frobenius symbols.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-022-00615-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10122623","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 Proof of the (frac{n!}{2}) Conjecture for Hook Shapes 关于钩形的$$frac{n!}{2}$$猜想的证明
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-10-20 DOI: 10.1007/s00026-022-00613-3
Sam Armon
{"title":"A Proof of the (frac{n!}{2}) Conjecture for Hook Shapes","authors":"Sam Armon","doi":"10.1007/s00026-022-00613-3","DOIUrl":"10.1007/s00026-022-00613-3","url":null,"abstract":"<div><p>A well-known representation-theoretic model for the transformed Macdonald polynomial <span>({widetilde{H}}_mu (Z;t,q))</span>, where <span>(mu )</span> is an integer partition, is given by the Garsia–Haiman module <span>({mathcal {H}}_mu )</span>. We study the <span>(frac{n!}{k})</span> conjecture of Bergeron and Garsia, which concerns the behavior of certain <i>k</i>-tuples of Garsia–Haiman modules under intersection. In the special case that <span>(mu )</span> has hook shape, we use a basis for <span>({mathcal {H}}_mu )</span> due to Adin, Remmel, and Roichman to resolve the <span>(frac{n!}{2})</span> conjecture by constructing an explicit basis for the intersection of two Garsia–Haiman modules.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-022-00613-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42687110","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
An Analogue of Mahonian Numbers and Log-Concavity Mahonian数与对数凹性的类比
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-10-18 DOI: 10.1007/s00026-022-00614-2
Yousra Ghemit, Moussa Ahmia
{"title":"An Analogue of Mahonian Numbers and Log-Concavity","authors":"Yousra Ghemit,&nbsp;Moussa Ahmia","doi":"10.1007/s00026-022-00614-2","DOIUrl":"10.1007/s00026-022-00614-2","url":null,"abstract":"<div><p>In this paper, we propose a <i>q</i>-analogue of the number of permutations <i>i</i>(<i>n</i>, <i>k</i>) of length <i>n</i> having <i>k</i> inversions known by Mahonian numbers. We investigate useful properties and some combinatorial interpretations by lattice paths/partitions and tilings. Furthermore, we give two constructive proofs of the strong <i>q</i>-log-concavity of the <i>q</i>-Mahonian numbers in <i>k</i> and <i>n</i>, respectively. In particular for <span>(q=1)</span>, we obtain two constructive proofs of the log-concavity of the Mahonian numbers in <i>k</i> and <i>n</i>, respectively.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-022-00614-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46634505","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
Symmetric Set Coloring of Signed Graphs 有符号图的对称集着色
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-10-15 DOI: 10.1007/s00026-022-00593-4
Chiara Cappello, Eckhard Steffen
{"title":"Symmetric Set Coloring of Signed Graphs","authors":"Chiara Cappello,&nbsp;Eckhard Steffen","doi":"10.1007/s00026-022-00593-4","DOIUrl":"10.1007/s00026-022-00593-4","url":null,"abstract":"<div><p>There are many concepts of signed graph coloring which are defined by assigning colors to the vertices of the graphs. These concepts usually differ in the number of self-inverse colors used. We introduce a unifying concept for this kind of coloring by assigning elements from symmetric sets to the vertices of the signed graphs. In the first part of the paper, we study colorings with elements from symmetric sets where the number of self-inverse elements is fixed. We prove a Brooks’-type theorem and upper bounds for the corresponding chromatic numbers in terms of the chromatic number of the underlying graph. These results are used in the second part where we introduce the symset-chromatic number <span>(chi _mathrm{sym}(G,sigma ))</span> of a signed graph <span>((G,sigma ))</span>. We show that the symset-chromatic number gives the minimum partition of a signed graph into independent sets and non-bipartite antibalanced subgraphs. In particular, <span>(chi _mathrm{sym}(G,sigma ) le chi (G))</span>. In the final section we show that these colorings can also be formalized as <i>DP</i>-colorings.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-022-00593-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47417762","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
Copartitions 合作
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-10-15 DOI: 10.1007/s00026-022-00607-1
Hannah E. Burson, Dennis Eichhorn
{"title":"Copartitions","authors":"Hannah E. Burson,&nbsp;Dennis Eichhorn","doi":"10.1007/s00026-022-00607-1","DOIUrl":"10.1007/s00026-022-00607-1","url":null,"abstract":"<div><p>We develop the theory of copartitions, which are a generalization of partitions with connections to many classical topics in partition theory, including Rogers–Ramanujan partitions, theta functions, mock theta functions, partitions with parts separated by parity, and crank statistics. Using both analytic and combinatorial methods, we give two forms of the three-parameter generating function, and we study several special cases that demonstrate the potential broader impact the study of copartitions may have.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45813576","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 Dumont Ansatz for the Eulerian Polynomials, Peak Polynomials and Derivative Polynomials Eulerian多项式、Peak多项式和导数多项式的Dumont-Ansatz
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2022-10-08 DOI: 10.1007/s00026-022-00609-z
William Y. C. Chen, Amy M. Fu
{"title":"The Dumont Ansatz for the Eulerian Polynomials, Peak Polynomials and Derivative Polynomials","authors":"William Y. C. Chen,&nbsp;Amy M. Fu","doi":"10.1007/s00026-022-00609-z","DOIUrl":"10.1007/s00026-022-00609-z","url":null,"abstract":"<div><p>We observe that three context-free grammars of Dumont can be brought to a common ground, via the idea of transformations of grammars, proposed by Ma–Ma–Yeh. Then we develop a unified perspective to investigate several combinatorial objects in connection with the bivariate Eulerian polynomials. We call this approach the Dumont ansatz. As applications, we provide grammatical treatments, in the spirit of the symbolic method, of relations on the Springer numbers, the Euler numbers, the three kinds of peak polynomials, an identity of Petersen, and the two kinds of derivative polynomials, introduced by Knuth–Buckholtz and Carlitz–Scoville, and later by Hoffman in a broader context. We obtain a convolution formula on the left peak polynomials, leading to the Gessel formula. In this framework, we come to the combinatorial interpretations of the derivative polynomials due to Josuat-Vergès.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44943966","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: 3
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