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Involution matrix loci and orbit harmonics 卷积矩阵位置和轨道谐波
arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06175
Moxuan J. Liu, Yichen Ma, Brendon Rhoades, Hai Zhu
{"title":"Involution matrix loci and orbit harmonics","authors":"Moxuan J. Liu, Yichen Ma, Brendon Rhoades, Hai Zhu","doi":"arxiv-2409.06175","DOIUrl":"https://doi.org/arxiv-2409.06175","url":null,"abstract":"Let $mathrm{Mat}_{n times n}(mathbb{C})$ be the affine space of $n times\u0000n$ complex matrices with coordinate ring $mathbb{C}[mathbf{x}_{n times n}]$.\u0000We define graded quotients of $mathbb{C}[mathbf{x}_{n times n}]$ which carry\u0000an action of the symmetric group $mathfrak{S}_n$ by simultaneous permutation\u0000of rows and columns. These quotient rings are obtained by applying the orbit\u0000harmonics method to matrix loci corresponding to all involutions in\u0000$mathfrak{S}_n$ and the conjugacy classes of involutions in $mathfrak{S}_n$\u0000with a given number of fixed points. In the case of perfect matchings on ${1,\u0000dots, n}$ with $n$ even, the Hilbert series of our quotient ring is related\u0000to Tracy-Widom distributions and its graded Frobenius image gives a refinement\u0000of the plethysm $s_{n/2}[s_2]$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"2013 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197632","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
Deformed Homogeneous $(s,t)$-Rogers-Szegö Polynomials and the Deformed $(s,t)$-Exponential Operator e$_{s,t}(y{rm T}_a D_{s,t},v)$ 变形均质 $(s,t)$ 罗格斯-席格多项式与变形 $(s,t)$ 指数算子 e$_{s,t}(y{rm T}_a D_{s,t},v)$
arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06878
Ronald Orozco López
{"title":"Deformed Homogeneous $(s,t)$-Rogers-Szegö Polynomials and the Deformed $(s,t)$-Exponential Operator e$_{s,t}(y{rm T}_a D_{s,t},v)$","authors":"Ronald Orozco López","doi":"arxiv-2409.06878","DOIUrl":"https://doi.org/arxiv-2409.06878","url":null,"abstract":"This article introduces the deformed homogeneous $(s,t)$-Rogers-Szeg\"o\u0000polynomials h$_{n}(x,y;s,t,u,v)$. These polynomials are a generalization of the\u0000Rogers-Szeg\"o polynomials and the $(p,q)$-Rogers-Szeg\"o polynomials defined\u0000by Jagannathan. By using the deformed $(s,t)$-exponential operator based on\u0000operator T$_{a}D_{s,t}$ we find identities involving the polynomials\u0000h$_{n}(x,y;s,t,u,v)$, together with generalizations of the Mehler and Rogers\u0000formulas. In addition, a generating function for the polynomials\u0000h$_{n}(x,y;s,t,u,v)$ is found employing the deformed\u0000$frac{varphi}{u}$-commuting operators. A representation of deformed\u0000$(s,t)$-exponential function as the limit of a sequence of deformed\u0000$(s,t)$-Rogers-Szeg\"o polynomials is obtained.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197610","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 number of H-free hypergraphs 关于无 H 超图的数量
arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06810
Tao Jiang, Sean Longbrake
{"title":"On the number of H-free hypergraphs","authors":"Tao Jiang, Sean Longbrake","doi":"arxiv-2409.06810","DOIUrl":"https://doi.org/arxiv-2409.06810","url":null,"abstract":"Two central problems in extremal combinatorics are concerned with estimating\u0000the number $ex(n,H)$, the size of the largest $H$-free hypergraph on $n$\u0000vertices, and the number $forb(n,H)$ of $H$-free hypergraph on $n$ vertices.\u0000While it is known that $forb(n,H)=2^{(1+o(1))ex(n,H)}$ for $k$-uniform\u0000hypergraphs that are not $k$-partite, estimates for hypergraphs that are\u0000$k$-partite (or degenerate) are not nearly as tight. In a recent breakthrough, Ferber, McKinley, and Samotij proved that for many\u0000degenerate hypergraphs $H$, $forb(n, H) = 2^{O(ex(n,H))}$. However, there are\u0000few known instances of degenerate hypergraphs $H$ for which\u0000$forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds. In this paper, we show that $forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds for a wide\u0000class of degenerate hypergraphs known as $2$-contractible hypertrees. This is\u0000the first known infinite family of degenerate hypergraphs $H$ for which\u0000$forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds. As a corollary of our main results, we\u0000obtain a surprisingly sharp estimate of\u0000$forb(n,C^{(k)}_ell)=2^{(lfloorfrac{ell-1}{2}rfloor+o(1))binom{n}{k-1}}$\u0000for the $k$-uniform linear $ell$-cycle, for all pairs $kgeq 5, ellgeq 3$,\u0000thus settling a question of Balogh, Narayanan, and Skokan affirmatively for all\u0000$kgeq 5, ellgeq 3$. Our methods also lead to some related sharp results on\u0000the corresponding random Turan problem. As a key ingredient of our proofs, we develop a novel supersaturation variant\u0000of the delta systems method for set systems, which may be of independent\u0000interest.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197613","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
The Converse of the Real Orthogonal Holant Theorem 实正交全等定理的逆定理
arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06911
Ben Young
{"title":"The Converse of the Real Orthogonal Holant Theorem","authors":"Ben Young","doi":"arxiv-2409.06911","DOIUrl":"https://doi.org/arxiv-2409.06911","url":null,"abstract":"The Holant theorem is a powerful tool for studying the computational\u0000complexity of counting problems in the Holant framework. Due to the great\u0000expressiveness of the Holant framework, a converse to the Holant theorem would\u0000itself be a very powerful counting indistinguishability theorem. The most\u0000general converse does not hold, but we prove the following, still highly\u0000general, version: if any two sets of real-valued signatures are\u0000Holant-indistinguishable, then they are equivalent up to an orthogonal\u0000transformation. This resolves a partially open conjecture of Xia (2010).\u0000Consequences of this theorem include the well-known result that homomorphism\u0000counts from all graphs determine a graph up to isomorphism, the classical\u0000sufficient condition for simultaneous orthogonal similarity of sets of real\u0000matrices, and a combinatorial characterization of simultaneosly orthogonally\u0000decomposable (odeco) sets of tensors.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197629","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
The asymptotic repetition threshold of sequences rich in palindromes 富含回文的序列的渐近重复阈值
arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06849
Lubomíra Dvořáková, Karel Klouda, Edita Pelantová
{"title":"The asymptotic repetition threshold of sequences rich in palindromes","authors":"Lubomíra Dvořáková, Karel Klouda, Edita Pelantová","doi":"arxiv-2409.06849","DOIUrl":"https://doi.org/arxiv-2409.06849","url":null,"abstract":"The asymptotic critical exponent measures for a sequence the maximum\u0000repetition rate of factors of growing length. The infimum of asymptotic\u0000critical exponents of sequences of a certain class is called the asymptotic\u0000repetition threshold of that class. On the one hand, if we consider the class\u0000of all d-ary sequences with d greater than one, then the asymptotic repetition\u0000threshold is equal to one, independently of the alphabet size. On the other\u0000hand, for the class of episturmian sequences, the repetition threshold depends\u0000on the alphabet size. We focus on rich sequences, i.e., sequences whose factors\u0000contain the maximum possible number of distinct palindromes. The class of\u0000episturmian sequences forms a subclass of rich sequences. We prove that the\u0000asymptotic repetition threshold for the class of rich recurrent d-ary\u0000sequences, with d greater than one, is equal to two, independently of the\u0000alphabet size.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225146","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 joint embedding property for cographs and trees 关于 cographs 和树的联合嵌入特性
arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06127
Daniel Carter
{"title":"On the joint embedding property for cographs and trees","authors":"Daniel Carter","doi":"arxiv-2409.06127","DOIUrl":"https://doi.org/arxiv-2409.06127","url":null,"abstract":"A family of graphs $mathcal{F}$ is said to have the joint embedding property\u0000(JEP) if for every $G_1, G_2in mathcal{F}$, there is an $Hin mathcal{F}$\u0000that contains both $G_1$ and $G_2$ as induced subgraphs. If $mathcal{F}$ is\u0000given by a finite set $S$ of forbidden induced subgraphs, it is known that\u0000determining if $mathcal{F}$ has JEP is undecidable. We prove that this problem\u0000is decidable if $P_4in S$ and generalize this result to families of rooted\u0000labeled trees under topological containment, bounded treewidth families under\u0000the graph minor relation, and bounded cliquewidth families under the induced\u0000subgraph relation.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197634","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
New bounds for the optimal density of covering single-insertion codes via the Turán density 通过图兰密度计算覆盖单插码最优密度的新界限
arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06425
Oleg Pikhurko, Oleg Verbitsky, Maksim Zhukovskii
{"title":"New bounds for the optimal density of covering single-insertion codes via the Turán density","authors":"Oleg Pikhurko, Oleg Verbitsky, Maksim Zhukovskii","doi":"arxiv-2409.06425","DOIUrl":"https://doi.org/arxiv-2409.06425","url":null,"abstract":"We prove that the density of any covering single-insertion code $Csubseteq\u0000X^r$ over the $n$-symbol alphabet $X$ cannot be smaller than $1/r+delta_r$ for\u0000some positive real $delta_r$ not depending on $n$. This improves the volume\u0000lower bound of $1/(r+1)$. On the other hand, we observe that, for all\u0000sufficiently large $r$, if $n$ tends to infinity then the asymptotic upper\u0000bound of $7/(r+1)$ due to Lenz et al (2021) can be improved to $4.911/(r+1)$. Both the lower and the upper bounds are achieved by relating the code density\u0000to the Tur'an density from extremal combinatorics. For the last task, we use\u0000the analytic framework of measurable subsets of the real cube $[0,1]^r$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225152","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 Systematic Approach to Crossing Numbers of Cartesian Products with Paths 直角坐标积与路径交叉数的系统方法
arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06755
Zayed Asiri, Ryan Burdett, Markus Chimani, Michael Haythorpe, Alex Newcombe, Mirko H. Wagner
{"title":"A Systematic Approach to Crossing Numbers of Cartesian Products with Paths","authors":"Zayed Asiri, Ryan Burdett, Markus Chimani, Michael Haythorpe, Alex Newcombe, Mirko H. Wagner","doi":"arxiv-2409.06755","DOIUrl":"https://doi.org/arxiv-2409.06755","url":null,"abstract":"Determining the crossing numbers of Cartesian products of small graphs with\u0000arbitrarily large paths has been an ongoing topic of research since the 1970s.\u0000Doing so requires the establishment of coincident upper and lower bounds; the\u0000former is usually demonstrated by providing a suitable drawing procedure, while\u0000the latter often requires substantial theoretical arguments. Many such papers\u0000have been published, which typically focus on just one or two small graphs at a\u0000time, and use ad hoc arguments specific to those graphs. We propose a general\u0000approach which, when successful, establishes the required lower bound. This\u0000approach can be applied to the Cartesian product of any graph with arbitrarily\u0000large paths, and in each case involves solving a modified version of the\u0000crossing number problem on a finite number (typically only two or three) of\u0000small graphs. We demonstrate the potency of this approach by applying it to\u0000Cartesian products involving all 133 graphs $G$ of orders five or six, and show\u0000that it is successful in 128 cases. This includes 60 cases which a recent\u0000survey listed as either undetermined, or determined only in journals without\u0000adequate peer review.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197611","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
Hamilton cycles in vertex-transitive graphs of order $6p$ 阶$6p$顶点变换图中的汉密尔顿循环
arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06138
Shaofei Du, Tianlei Zhou
{"title":"Hamilton cycles in vertex-transitive graphs of order $6p$","authors":"Shaofei Du, Tianlei Zhou","doi":"arxiv-2409.06138","DOIUrl":"https://doi.org/arxiv-2409.06138","url":null,"abstract":"It was shown that every connected vertex-transitive graph of order $6p$,\u0000where $p$ is a prime, contains a Hamilton path cite{KS09}. It will be shown in\u0000this paper that every such graph contains a Hamilton cycle, except for the\u0000triangle-replaced graph of the Petersen graph.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197633","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
Generalized Diagonals in Positive Semi-Definite Matrices 正半有限矩阵中的广义对角线
arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06907
Robert Angarone, Daniel Soskin
{"title":"Generalized Diagonals in Positive Semi-Definite Matrices","authors":"Robert Angarone, Daniel Soskin","doi":"arxiv-2409.06907","DOIUrl":"https://doi.org/arxiv-2409.06907","url":null,"abstract":"We describe all inequalities among generalized diagonals in positive\u0000semi-definite matrices. These turn out to be governed by a simple partial order\u0000on the symmetric group. This provides an analogue of results of Drake, Gerrish,\u0000and Skandera on inequalities among generalized diagonals in totally nonnegative\u0000matrices.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"206 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197608","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
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