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ON THE EXPLICIT FORMULA FOR GAUSS-JORDAN ELIMINATION 高斯约当消去的显式公式
arXiv: Combinatorics Pub Date : 2020-10-02 DOI: 10.17654/nt048020155
N. Tran, Júlia Justino, I. Berg
{"title":"ON THE EXPLICIT FORMULA FOR GAUSS-JORDAN ELIMINATION","authors":"N. Tran, Júlia Justino, I. Berg","doi":"10.17654/nt048020155","DOIUrl":"https://doi.org/10.17654/nt048020155","url":null,"abstract":"The elements of the successive intermediate matrices of the Gauss-Jordan elimination procedure have the form of quotients of minors. Instead of the proof using identities of determinants of cite{Li}, a direct proof by induction is given.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89586665","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
The Intersection Spectrum of 3-Chromatic Intersecting Hypergraphs 三色相交超图的相交谱
arXiv: Combinatorics Pub Date : 2020-10-01 DOI: 10.1007/978-3-030-83823-2_22
M. Buci'c, Stefan Glock, B. Sudakov
{"title":"The Intersection Spectrum of 3-Chromatic Intersecting Hypergraphs","authors":"M. Buci'c, Stefan Glock, B. Sudakov","doi":"10.1007/978-3-030-83823-2_22","DOIUrl":"https://doi.org/10.1007/978-3-030-83823-2_22","url":null,"abstract":"","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80043302","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
On some problems about ternary paths: a linear algebra approach 关于三元路径的若干问题:线性代数方法
arXiv: Combinatorics Pub Date : 2020-09-29 DOI: 10.1216/rmj.2021.51.709
H. Prodinger
{"title":"On some problems about ternary paths: a linear algebra approach","authors":"H. Prodinger","doi":"10.1216/rmj.2021.51.709","DOIUrl":"https://doi.org/10.1216/rmj.2021.51.709","url":null,"abstract":"Ternary paths consist of an up-step of one unit, a down-step of two units, never go below the $x$-axis, and return to the $x$-axis. This paper addresses the enumeration of partial ternary paths, ending at a given level $i$, reading the path either from left to right or from right to left. Since the paths are not symmetric w.r.t. left vs. right, as classical Dyck paths, this leads to different results. The right to left enumeration is quite challenging, but leads at the end to very satisfying results. The methods are elementary (solving systems of linear equations). In this way, several conjectures left open in Naiomi Cameron's Ph.D. thesis could be successfully settled.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89857826","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
The Primitive Derivation and Discrete Integrals 原始导数与离散积分
arXiv: Combinatorics Pub Date : 2020-09-29 DOI: 10.3842/SIGMA.2021.038
M. Yoshinaga, D. Suyama
{"title":"The Primitive Derivation and Discrete Integrals","authors":"M. Yoshinaga, D. Suyama","doi":"10.3842/SIGMA.2021.038","DOIUrl":"https://doi.org/10.3842/SIGMA.2021.038","url":null,"abstract":"The modules of logarithmic derivations for the (extended) Catalan and Shi arrangements associated with root systems are known to be free. However, except for a few cases, explicit bases for such modules are not known. In this paper, we construct explicit bases for type $A$ root systems. \u0000Our construction is based on Bandlow-Musiker's integral formula for a basis of the space of quasiinvariants. The integral formula can be considered as an expression for the inverse of the primitive derivation introduced by K. Saito. We prove that the discrete analogues of the integral formulas provide bases for Catalan and Shi arrangements.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83033772","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
Expansion in Matrix-Weighted Graphs 矩阵加权图的展开
arXiv: Combinatorics Pub Date : 2020-09-25 DOI: 10.1016/J.LAA.2021.08.009
J. Hansen
{"title":"Expansion in Matrix-Weighted Graphs","authors":"J. Hansen","doi":"10.1016/J.LAA.2021.08.009","DOIUrl":"https://doi.org/10.1016/J.LAA.2021.08.009","url":null,"abstract":"","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83965775","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: 3
Laplacian integral graphs with a given degreee sequence constraint 给定次序列约束的拉普拉斯积分图
arXiv: Combinatorics Pub Date : 2020-09-24 DOI: 10.22199/issn.0717-6279-4753
A. Novanta, C. Oliveira, L. de Lima
{"title":"Laplacian integral graphs with a given degreee sequence constraint","authors":"A. Novanta, C. Oliveira, L. de Lima","doi":"10.22199/issn.0717-6279-4753","DOIUrl":"https://doi.org/10.22199/issn.0717-6279-4753","url":null,"abstract":"Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) - A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral is all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all L-integral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84718829","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
On the Powers of Signed Graphs 论符号图的幂
arXiv: Combinatorics Pub Date : 2020-09-22 DOI: 10.22049/CCO.2021.27083.1193
T. Shijin, A. GerminaK., K. ShahulHameed
{"title":"On the Powers of Signed Graphs","authors":"T. Shijin, A. GerminaK., K. ShahulHameed","doi":"10.22049/CCO.2021.27083.1193","DOIUrl":"https://doi.org/10.22049/CCO.2021.27083.1193","url":null,"abstract":"A signed graph is an ordered pair $Sigma=(G,sigma),$ where $G=(V,E)$ is the underlying graph of $Sigma$ with a signature function $sigma:Erightarrow {1,-1}$. In this article, we define $n^{th}$ power of a signed graph and discuss some properties of these powers of signed graphs. As we can define two types of signed graphs as the power of a signed graph, necessary and sufficient conditions are given for an $n^{th}$ power of a signed graph to be unique. Also, we characterize balanced power signed graphs.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"39 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72591411","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
A symmetric function generalization of the Zeilberger–Bressoud $q$-Dyson theorem Zeilberger-Bressoud $q$-Dyson定理的对称函数推广
arXiv: Combinatorics Pub Date : 2020-09-10 DOI: 10.1090/proc/15399
Yue Zhou
{"title":"A symmetric function generalization of the Zeilberger–Bressoud $q$-Dyson theorem","authors":"Yue Zhou","doi":"10.1090/proc/15399","DOIUrl":"https://doi.org/10.1090/proc/15399","url":null,"abstract":"In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson theorem or the $q$-Dyson constant term identity. This conjecture was proved by Karolyi, Lascoux and Warnaar in 2015. In this paper, by slightly changing the variables of Kadell's conjecture, we obtain another symmetric function generalization of the $q$-Dyson constant term identity. This new generalized constant term admits a simple product-form expression.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"25 4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77389157","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
Higher Order Apostol-Type Poly-Genocchi Polynomials with Parameters a, b and c 具有参数a, b和c的高阶apostoltype多- genochi多项式
arXiv: Combinatorics Pub Date : 2020-09-08 DOI: 10.4134/CKMS.C200275
C. Corcino, R. Corcino
{"title":"Higher Order Apostol-Type Poly-Genocchi Polynomials with Parameters a, b and c","authors":"C. Corcino, R. Corcino","doi":"10.4134/CKMS.C200275","DOIUrl":"https://doi.org/10.4134/CKMS.C200275","url":null,"abstract":"In this paper, a new form of poly-Genocchi polynomials is defined by means of poly-logarithm, namely, the Apostol-type poly-Genocchi polynomials of higher order with parameters a, b and c. Several properties of these polynomials are established including some recurrence relations and explicit formulas, which express these higher order Apostol-type poly-Genocchi polynomials in terms of Stirling numbers of the second kind, Apostol-type Bernoulli and Frobenius polynomials of higher order. Moreover, certain differential identity is obtained that leads this new form of poly-Genocchi polynomials to be classified as Appell polynomials and, consequently, draw more properties using some theorems on Appell polynomials. Furthermore, a symmetrized generalization of this new form of poly-Genocchi polynomials is introduced that possesses a double generating function. Finally, the type 2 Apostol-poly-Genocchi polynomials with parameters a, b and c are defined using the concept of polyexponential function and several identities are derived, two of which show the connections of these polynomials with Stirling numbers of the first kind and the type 2 Apostol-type poly-Bernoulli polynomials.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78444442","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
The Extremal Number of Tight Cycles 紧循环的极值数
arXiv: Combinatorics Pub Date : 2020-09-01 DOI: 10.1093/IMRN/RNAA396
B. Sudakov, István Tomon
{"title":"The Extremal Number of Tight Cycles","authors":"B. Sudakov, István Tomon","doi":"10.1093/IMRN/RNAA396","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA396","url":null,"abstract":"A tight cycle in an $r$-uniform hypergraph $mathcal{H}$ is a sequence of $ellgeq r+1$ vertices $x_1,dots,x_{ell}$ such that all $r$-tuples ${x_{i},x_{i+1},dots,x_{i+r-1}}$ (with subscripts modulo $ell$) are edges of $mathcal{H}$. \u0000An old problem of V. Sos, also posed independently by J. Verstraete, asks for the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices which has no tight cycle. Although this is a very basic question, until recently, no good upper bounds were known for this problem for $rgeq 3$. Here we prove that the answer is at most $n^{r-1+o(1)}$, which is tight up to the $o(1)$ error term. \u0000Our proof is based on finding robust expanders in the line graph of $mathcal{H}$ together with certain density increment type arguments.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"125 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73948626","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: 15
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