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arXiv: Combinatorics最新文献

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Classification of indecomposable 2 r -divisible codes spanned by by codewords of weight 2 r 权值为2r的码字所组成的不可分解2r可分码的分类
arXiv: Combinatorics Pub Date : 2020-11-11 DOI: 10.15495/EPUB_UBT_00005157
Sascha Kurz
{"title":"Classification of indecomposable 2 r -divisible codes spanned by by codewords of weight 2 r","authors":"Sascha Kurz","doi":"10.15495/EPUB_UBT_00005157","DOIUrl":"https://doi.org/10.15495/EPUB_UBT_00005157","url":null,"abstract":"We classify indecomposable binary linear codes whose weights of the codewords are divisible by $2^r$ for some integer $r$ and that are spanned by the set of minimum weight codewords.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83316259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Logarithmic Generalization of the Lambert W Function and Its Applications to Adiabatic Thermostatistics of the Three-Parameter Entropy Lambert W函数的对数推广及其在三参数熵绝热统计中的应用
arXiv: Combinatorics Pub Date : 2020-11-06 DOI: 10.1155/2021/6695559
C. Corcino, R. Corcino
{"title":"Logarithmic Generalization of the Lambert \u0000 W\u0000 Function and Its Applications to Adiabatic Thermostatistics of the Three-Parameter Entropy","authors":"C. Corcino, R. Corcino","doi":"10.1155/2021/6695559","DOIUrl":"https://doi.org/10.1155/2021/6695559","url":null,"abstract":"A generalization of the Lambert W function called the logarithmic Lambert function is found to be a solution to the thermostatics of the three-parameter entropy of classical ideal gas in adiabatic ensembles. The derivative, integral, Taylor series, approximation formula and branches of the function are obtained. The thermostatics are computed and the heat functions are expressed in terms of the logarithmic Lambert function.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86820409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Semi-invariants of binary forms pertaining to a unimodality theorem of Reiner and Stanton 二元形式的半不变量与Reiner和Stanton的单模定理有关
arXiv: Combinatorics Pub Date : 2020-11-03 DOI: 10.1142/S0129167X21400036
William Y. C. Chen, I. D. Jia
{"title":"Semi-invariants of binary forms pertaining to a unimodality theorem of Reiner and Stanton","authors":"William Y. C. Chen, I. D. Jia","doi":"10.1142/S0129167X21400036","DOIUrl":"https://doi.org/10.1142/S0129167X21400036","url":null,"abstract":"The strange symmetric difference of the $q$-binomial coefficients $F_{n,k}(q)={n+kbrack k}-q^{n}{n+k-2brack k-2}$ as called by Stanley, was introduced by Reiner and Stanton. They proved that $F_{n,k}(q)$ is symmetric and unimodal for $kgeq 2$ and any even nonnegative integer $n$ by using the representation theory for Lie algebras. Inspired by the connection between the Gaussian coefficients, or the $q$-binomial coefficients, and semi-invariants of binary forms established by Sylvester in his proof of the unimodality of the Gaussian coefficients as conjectured by Cayley, we find an interpretation of the unimodality of $F_{n,k}(q)$ in terms of semi-invariants. In the spirit of the strict unimodality of the Gaussian coefficients due to Pak and Panova, we prove the strict unimodality of $G_{n,k,r}(q)={n+kbrack k}-q^{nr/2}{n+k-rbrack k-r}$, where $n,rgeq8$, $kgeq r$ and either $n$ or $r$ is even.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86659719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Trees and cycles. 树木和自行车。
arXiv: Combinatorics Pub Date : 2020-10-28 DOI: 10.1201/9781315272689-4
P. Cameron, Liam Stott
{"title":"Trees and cycles.","authors":"P. Cameron, Liam Stott","doi":"10.1201/9781315272689-4","DOIUrl":"https://doi.org/10.1201/9781315272689-4","url":null,"abstract":"Let $T$ be a tree on $n$ vertices. We can regard the edges of $T$ as transpositions of the vertex set; their product (in any order) is a cyclic permutation. All possible cyclic permutations arise (each exactly once) if and only if the tree is a star. In this paper we find the number of realised cycles, and obtain some results on the number of realisations of each cycle, for other trees. We also solve the inverse problem of the number of trees which give rise to a given cycle. On the way, we meet some familiar number sequences including the Euler and Fuss--Catalan numbers.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84718821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Constructing dense grid-free linear 3-graphs 构造密集无网格线性3图
arXiv: Combinatorics Pub Date : 2020-10-27 DOI: 10.1090/PROC/15673
Lior Gishboliner, A. Shapira
{"title":"Constructing dense grid-free linear 3-graphs","authors":"Lior Gishboliner, A. Shapira","doi":"10.1090/PROC/15673","DOIUrl":"https://doi.org/10.1090/PROC/15673","url":null,"abstract":"We show that there exist linear $3$-uniform hypergraphs with $n$ vertices and $Omega(n^2)$ edges which contain no copy of the $3 times 3$ grid. This makes significant progress on a conjecture of F\"{u}redi and Ruszink'{o}. We also discuss connections to proving lower bounds for the $(9,6)$ Brown-ErdH{o}s-S'{o}s problem and to a problem of Solymosi and Solymosi.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78040472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
On the maximum number of minimal codewords 最小码字的最大数目
arXiv: Combinatorics Pub Date : 2020-10-21 DOI: 10.15495/EPUB_UBT_00005128
R. D. Cruz, Sascha Kurz
{"title":"On the maximum number of minimal codewords","authors":"R. D. Cruz, Sascha Kurz","doi":"10.15495/EPUB_UBT_00005128","DOIUrl":"https://doi.org/10.15495/EPUB_UBT_00005128","url":null,"abstract":"We study the minimum number of minimal codewords in linear codes from the point of view of projective geometry. We derive bounds and in some cases determine the exact values. We also present an extension to minimal subcode supports.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73769308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Algebraic methods in difference sets and bent functions 差分集和弯曲函数的代数方法
arXiv: Combinatorics Pub Date : 2020-10-17 DOI: 10.13069/JACODESMATH.940192
P. H. Keskar, P. Kumari
{"title":"Algebraic methods in difference sets and bent functions","authors":"P. H. Keskar, P. Kumari","doi":"10.13069/JACODESMATH.940192","DOIUrl":"https://doi.org/10.13069/JACODESMATH.940192","url":null,"abstract":"We provide some applications of a polynomial criterion for difference sets. These include counting the difference sets with specified parameters in terms of Hilbert functions, in particular a count of bent functions. We also consider the question about the bentness of certain Boolean functions introduced by Carlet when the $mathcal{C}$-condition introduced by him doesn't hold.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79872943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Symmetric polynomials associated with numerical semigroups 与数值半群相关的对称多项式
arXiv: Combinatorics Pub Date : 2020-10-07 DOI: 10.47443/dml.2020.0066
L. Fel
{"title":"Symmetric polynomials associated with numerical semigroups","authors":"L. Fel","doi":"10.47443/dml.2020.0066","DOIUrl":"https://doi.org/10.47443/dml.2020.0066","url":null,"abstract":"We study a new kind of symmetric polynomials P_n(x_1,...,x_m) of degree n in m real variables, which have arisen in the theory of numerical semigroups. We establish their basic properties and find their representation through the power sums E_k=sum_{j=1}^m x_j^k. We observe a visual similarity between normalized polynomials P_n(x_1,...,x_m)/chi_m, where chi_m=prod_{j=1}^m x_j, and a polynomial part of a partition function W(s,{d_1,...,d_m}), which gives a number of partitions of sge 0 into m positive integers d_j, and put forward a conjecture about their relationship.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78176458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Excellent graphs with respect to domination: subgraphs induced by minimum dominating sets 关于支配的优秀图:由最小支配集诱导的子图
arXiv: Combinatorics Pub Date : 2020-10-07 DOI: 10.47443/dml.2020.0052
V. Samodivkin
{"title":"Excellent graphs with respect to domination: subgraphs induced by minimum dominating sets","authors":"V. Samodivkin","doi":"10.47443/dml.2020.0052","DOIUrl":"https://doi.org/10.47443/dml.2020.0052","url":null,"abstract":"A graph $G=(V,E)$ is $gamma$-excellent if $V$ is a union of all $gamma$-sets of $G$, where $gamma$ stands for the domination number. Let $mathcal{I}$ be a set of all mutually nonisomorphic graphs and $emptyset not= mathcal{H} subsetneq mathcal{I}$. In this paper we initiate the study of the $mathcal{H}$-$gamma$-excellent graphs, which we define as follows. A graph $G$ is $mathcal{H}$-$gamma$-excellent if the following hold: (i) for every $H in mathcal{H}$ and for each $x in V(G)$ there exists an induced subgraph $H_x$ of $G$ such that $H$ and $H_x$ are isomorphic, $x in V(H_x)$ and $V(H_x)$ is a subset of some $gamma$-set of $G$, and (ii) the vertex set of every induced subgraph $H$ of $G$, which is isomorphic to some element of $mathcal{H}$, is a subset of some $gamma$-set of $G$. For each of some well known graphs, including cycles, trees and some cartesian products of two graphs, we describe its largest set $mathcal{H} subsetneq mathcal{I}$ for which the graph is $mathcal{H}$-$gamma$-excellent. Results on $gamma$-excellent regular graphs and a generalized lexicographic product of graphs are presented. Several open problems and questions are posed.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73399991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Three new lengths for cyclic Legendre pairs 循环勒让德对的三个新长度
arXiv: Combinatorics Pub Date : 2020-10-06 DOI: 10.31799/1684-8853-2021-1-2-7
N. Balonin, Dragomiru Z. Dokovic
{"title":"Three new lengths for cyclic Legendre pairs","authors":"N. Balonin, Dragomiru Z. Dokovic","doi":"10.31799/1684-8853-2021-1-2-7","DOIUrl":"https://doi.org/10.31799/1684-8853-2021-1-2-7","url":null,"abstract":"There are 20 odd integers v less than 200 for which the existence of Legendre pairs of length v is undecided. The smallest among them is v=77. We have constructed Legendre pairs of lengths 91, 93 and 123 reducing the number of undecided cases to 17.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76821493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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