Pub Date : 2020-11-11DOI: 10.15495/EPUB_UBT_00005157
Sascha Kurz
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.
{"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":"27 1","pages":""},"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}
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.
{"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":"60 1","pages":""},"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}
Pub Date : 2020-11-03DOI: 10.1142/S0129167X21400036
William Y. C. Chen, I. D. Jia
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.
{"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":"94 1","pages":""},"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}
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.
{"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":"8 1","pages":""},"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}
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.
我们证明了存在具有$n$顶点和$Omega(n^2)$边的线性$3$-均匀超图，它不包含$3 乘以3$网格的副本。这在F {u}redi和Ruszink {o}的一个猜想上取得了重大进展。我们还讨论了$(9,6)$ Brown-ErdH{o} - s '{o}s问题下界的证明以及关于Solymosi和Solymosi问题的证明的联系。
{"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":"21 1","pages":""},"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}
Pub Date : 2020-10-21DOI: 10.15495/EPUB_UBT_00005128
R. D. Cruz, Sascha Kurz
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.
{"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":"13 1","pages":""},"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}
Pub Date : 2020-10-17DOI: 10.13069/JACODESMATH.940192
P. H. Keskar, P. Kumari
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.
{"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":"42 1","pages":""},"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}
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.
{"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":"136 1","pages":""},"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}
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.
图$G=(V,E)$是$gamma$-如果$V$是$G$的所有$gamma$集的并集，其中$gamma$表示支配数，则$ $ $-是优秀的。设$mathcal{I}$是所有相互非同构图的集合，且$emptyset not= mathcal{H} subsetneq mathcal{I}$。在本文中，我们开始研究$mathcal{H}$-$gamma$-优图，我们定义如下。图G是$美元 mathcal {H} - $ 伽马优秀美元如果以下:(i)每美元H mathcal {H}和为每个x美元 V (G)存在一个诱导子图H_x美元$ G H和美元,美元的H_x同构,美元$ x V (H_x)和V (H_x)美元是美元的一个子集伽马集G美元,美元和(2)每一个诱导子图的顶点集H G美元,美元是美元的同构某个元素 mathcal {H},美元是美元的子集伽马设定的G美元美元。对于每一个已知的图，包括环，树和一些两个图的笛卡尔积，我们描述了它的最大集$mathcal{H} subsetneq mathcal{I}$，其中图是$mathcal{H}$-$gamma$-excellent。给出了$gamma$-优正则图的结果和图的广义词典积。提出了几个悬而未决的问题。
{"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":"1 1","pages":""},"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}
Pub Date : 2020-10-06DOI: 10.31799/1684-8853-2021-1-2-7
N. Balonin, Dragomiru Z. Dokovic
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.
{"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":"4 1","pages":""},"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}
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