Pub Date : 2024-02-22DOI: 10.1016/j.jcta.2024.105874
Shane Chern , Ernest X.W. Xia
In their study of the truncation of Euler's pentagonal number theorem, Andrews and Merca introduced a partition function to count the number of partitions of n in which k is the least integer that is not a part and there are more parts exceeding k than there are below k. In recent years, two conjectures concerning on truncated theta series were posed by Andrews, Merca, and Yee. In this paper, we prove that the two conjectures are true for sufficiently large n whenever k is fixed.
安德鲁斯和梅尔卡在研究欧拉五边形数截断定理时,引入了一个分区函数 Mk(n),用来计算 n 的分区数,其中 k 是不属于分区的最小整数,且超过 k 的分区数多于低于 k 的分区数。近年来,安德鲁斯、梅尔卡和易提出了关于截断θ数列 Mk(n) 的两个猜想。在本文中,我们证明了只要 k 固定不变,对于足够大的 n,这两个猜想都是真的。
{"title":"Two conjectures of Andrews, Merca and Yee on truncated theta series","authors":"Shane Chern , Ernest X.W. Xia","doi":"10.1016/j.jcta.2024.105874","DOIUrl":"10.1016/j.jcta.2024.105874","url":null,"abstract":"<div><p>In their study of the truncation of Euler's pentagonal number theorem, Andrews and Merca introduced a partition function <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> to count the number of partitions of <em>n</em> in which <em>k</em> is the least integer that is not a part and there are more parts exceeding <em>k</em> than there are below <em>k</em>. In recent years, two conjectures concerning <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> on truncated theta series were posed by Andrews, Merca, and Yee. In this paper, we prove that the two conjectures are true for sufficiently large <em>n</em> whenever <em>k</em> is fixed.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105874"},"PeriodicalIF":1.1,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139937815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-21DOI: 10.1016/j.jcta.2024.105873
L. Mella , T. Traetta
<div><p>Let <em>S</em> be a subset of a group <em>G</em> (not necessarily abelian) such that <span><math><mi>S</mi><mspace></mspace><mo>∩</mo><mo>−</mo><mi>S</mi></math></span> is empty or contains only elements of order 2, and let <span><math><mi>h</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> and <span><math><mi>k</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. A <em>generalized Heffter array</em> GHA<span><math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> over <em>G</em> is an <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> matrix <span><math><mi>A</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></math></span> such that: the <em>i</em>-th row (resp. <em>j</em>-th column) of <em>A</em> contains exactly <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> (resp. <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>) nonzero elements, and the list <span><math><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≠</mo><mn>0</mn><mo>}</mo></math></span> equals <em>λ</em> times the set <span><math><mi>S</mi><mspace></mspace><mo>∪</mo><mspace></mspace><mo>−</mo><mi>S</mi></math></span>. We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of <em>A</em> sums to zero (resp. a nonzero element), with respect to some ordering.</p><p>In this paper, we use <em>near alternating sign matrices</em> to build both zero and nonzero sum GHAs, over cyclic groups, having the further strong property of being simple. In particular, we construct zero sum and simple GHAs whose row and column weights are congruent to 0 modulo 4. This result also provides the first infinite family of simple (classic) Heffter arrays to be rectangular (<span><math><mi>m</mi><mo>≠</mo><mi>n</mi></math></span>) and with less than <em>n</em> nonzero entries in each row. Furthermore, we build nonzero sum GHA<span><math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> over an arbitrary group <em>G</em> whenever <em>S</em> contains enough noninvolutions, th
设 S 是一个群 G 的子集(不一定是非良性的),使得 S∩-S 是空的或只包含阶数为 2 的元素,并且设 h=(h1,...,hm)∈Nm,k=(k1,...,kn )∈Nn。G 上的广义赫夫特数组 GHASλ(m,n;h,k) 是一个 m×n 矩阵 A=(aij),使得:A 的第 i 行(或第 j 列)恰好包含 hi(或 kj)个非零元素,且列表 {aij,-aij|aij≠0} 等于集合 S∪-S 的 λ 倍。在本文中,我们使用近交替符号矩阵来构建循环群上的零和(或非零和)GHA,它还具有简单的强性质。特别是,我们构建的零和简单 GHA,其行权重和列权重同余为 0 modulo 4。这一结果还提供了第一个矩形(m≠n)且每行非零条目少于 n 个的简单(经典)赫夫特数组无穷族。此外,只要 S 包含足够多的非卷积,我们就能在任意群 G 上建立非零和 GHASλ(m,n;h,k),从而扩展了之前的非构造性结果,即对于 G 的某个子群 H,±S=G∖H。最后,我们描述了如何利用 GHA 来建立 Cayley 图(在不一定是无性的群上)到可定向曲面上的正交分解和双嵌套。
{"title":"Constructing generalized Heffter arrays via near alternating sign matrices","authors":"L. Mella , T. Traetta","doi":"10.1016/j.jcta.2024.105873","DOIUrl":"10.1016/j.jcta.2024.105873","url":null,"abstract":"<div><p>Let <em>S</em> be a subset of a group <em>G</em> (not necessarily abelian) such that <span><math><mi>S</mi><mspace></mspace><mo>∩</mo><mo>−</mo><mi>S</mi></math></span> is empty or contains only elements of order 2, and let <span><math><mi>h</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> and <span><math><mi>k</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. A <em>generalized Heffter array</em> GHA<span><math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> over <em>G</em> is an <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> matrix <span><math><mi>A</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></math></span> such that: the <em>i</em>-th row (resp. <em>j</em>-th column) of <em>A</em> contains exactly <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> (resp. <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>) nonzero elements, and the list <span><math><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≠</mo><mn>0</mn><mo>}</mo></math></span> equals <em>λ</em> times the set <span><math><mi>S</mi><mspace></mspace><mo>∪</mo><mspace></mspace><mo>−</mo><mi>S</mi></math></span>. We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of <em>A</em> sums to zero (resp. a nonzero element), with respect to some ordering.</p><p>In this paper, we use <em>near alternating sign matrices</em> to build both zero and nonzero sum GHAs, over cyclic groups, having the further strong property of being simple. In particular, we construct zero sum and simple GHAs whose row and column weights are congruent to 0 modulo 4. This result also provides the first infinite family of simple (classic) Heffter arrays to be rectangular (<span><math><mi>m</mi><mo>≠</mo><mi>n</mi></math></span>) and with less than <em>n</em> nonzero entries in each row. Furthermore, we build nonzero sum GHA<span><math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> over an arbitrary group <em>G</em> whenever <em>S</em> contains enough noninvolutions, th","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105873"},"PeriodicalIF":1.1,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139916880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-14DOI: 10.1016/j.jcta.2024.105870
Francesco Fumagalli , Martino Garonzi , Pietro Gheri
Let G be the alternating group of degree n. Let be the maximal size of a subset S of G such that whenever and and let be the minimal size of a family of proper subgroups of G whose union is G. We prove that, when n varies in the family of composite numbers, tends to 1 as . Moreover, we explicitly calculate for congruent to 3 modulo 18.
设 G 是 n 阶交替群。设 ω(G) 是 G 的子集 S 的最大大小,当 x,y∈S 且 x≠y 时,使得〈x,y〉=G;设 σ(G) 是 G 的一族适当子群的最小大小,其联合是 G。我们证明,当 n 在合数族中变化时,σ(G)/ω(G) 随着 n→∞ 趋于 1。此外,我们还明确地计算了 n≥21 的 σ(An)与 3 的同余式 18 的同余式。
{"title":"On the maximal number of elements pairwise generating the finite alternating group","authors":"Francesco Fumagalli , Martino Garonzi , Pietro Gheri","doi":"10.1016/j.jcta.2024.105870","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105870","url":null,"abstract":"<div><p>Let <em>G</em> be the alternating group of degree <em>n</em>. Let <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the maximal size of a subset <em>S</em> of <em>G</em> such that <span><math><mo>〈</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>〉</mo><mo>=</mo><mi>G</mi></math></span> whenever <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi></math></span> and <span><math><mi>x</mi><mo>≠</mo><mi>y</mi></math></span> and let <span><math><mi>σ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the minimal size of a family of proper subgroups of <em>G</em> whose union is <em>G</em>. We prove that, when <em>n</em> varies in the family of composite numbers, <span><math><mi>σ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>/</mo><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> tends to 1 as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>. Moreover, we explicitly calculate <span><math><mi>σ</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>21</mn></math></span> congruent to 3 modulo 18.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105870"},"PeriodicalIF":1.1,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000098/pdfft?md5=0f713e308f01065a0eed53c25b2ba78c&pid=1-s2.0-S0097316524000098-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139732729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-09DOI: 10.1016/j.jcta.2024.105871
Shamil Asgarli , Dragos Ghioca , Chi Hoi Yip
A plane curve of degree d is called blocking if every -line in the plane meets C at some -point. We prove that the proportion of blocking curves among those of degree d is when and . We also show that the same conclusion holds for smooth curves under the somewhat weaker condition and . Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of -roots of random polynomials, we find that the limiting distribution of the number of -points in the intersection of a random plane curve and a fixed -line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of -points contained in a union of k lines for .
如果平面中的每条 Fq 线都与 C 交于某个 Fq 点,则阶数为 d 的平面曲线 C⊂P2 称为阻塞曲线。我们证明,当 d≥2q-1 且 q→∞ 时,阻塞曲线在 d 阶曲线中所占比例为 o(1)。我们还证明,在较弱的条件 d≥3p 和 d,q→∞ 下,同样的结论也适用于光滑曲线。此外,随机平面曲线的光滑和阻塞两种情况被证明是渐近独立的。我们扩展了关于随机多项式 Fq 根数的经典结果,发现随机平面曲线与固定 Fq 线交点的 Fq 点数的极限分布是均值为 1 的泊松分布。我们还给出了阻塞曲线比例的明确公式,其中涉及 k=1,2,...q2+q+1 时 k 线结合处所含 Fq 点数的统计量。
{"title":"Most plane curves over finite fields are not blocking","authors":"Shamil Asgarli , Dragos Ghioca , Chi Hoi Yip","doi":"10.1016/j.jcta.2024.105871","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105871","url":null,"abstract":"<div><p>A plane curve <span><math><mi>C</mi><mo>⊂</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> of degree <em>d</em> is called <em>blocking</em> if every <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-line in the plane meets <em>C</em> at some <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-point. We prove that the proportion of blocking curves among those of degree <em>d</em> is <span><math><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> when <span><math><mi>d</mi><mo>≥</mo><mn>2</mn><mi>q</mi><mo>−</mo><mn>1</mn></math></span> and <span><math><mi>q</mi><mo>→</mo><mo>∞</mo></math></span>. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition <span><math><mi>d</mi><mo>≥</mo><mn>3</mn><mi>p</mi></math></span> and <span><math><mi>d</mi><mo>,</mo><mi>q</mi><mo>→</mo><mo>∞</mo></math></span>. Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-roots of random polynomials, we find that the limiting distribution of the number of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-points in the intersection of a random plane curve and a fixed <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-points contained in a union of <em>k</em> lines for <span><math><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"204 ","pages":"Article 105871"},"PeriodicalIF":1.1,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139719451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-09DOI: 10.1016/j.jcta.2024.105872
Paul Terwilliger
<div><p>The goal of this article is to display a <em>Q</em>-polynomial structure for the Attenuated Space poset <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span>. The poset <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span> is briefly described as follows. Start with an <span><math><mo>(</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo>)</mo></math></span>-dimensional vector space <em>H</em> over a finite field with <em>q</em> elements. Fix an <em>M</em>-dimensional subspace <em>h</em> of <em>H</em>. The vertex set <em>X</em> of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span> consists of the subspaces of <em>H</em> that have zero intersection with <em>h</em>. The partial order on <em>X</em> is the inclusion relation. The <em>Q</em>-polynomial structure involves two matrices <span><math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>∈</mo><msub><mrow><mi>Mat</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> with the following entries. For <span><math><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>X</mi></math></span> the matrix <em>A</em> has <span><math><mo>(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></math></span>-entry 1 (if <em>y</em> covers <em>z</em>); <span><math><msup><mrow><mi>q</mi></mrow><mrow><mrow><mi>dim</mi></mrow><mspace></mspace><mi>y</mi></mrow></msup></math></span> (if <em>z</em> covers <em>y</em>); and 0 (if neither of <span><math><mi>y</mi><mo>,</mo><mi>z</mi></math></span> covers the other). The matrix <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is diagonal, with <span><math><mo>(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span>-entry <span><math><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mrow><mi>dim</mi></mrow><mspace></mspace><mi>y</mi></mrow></msup></math></span> for all <span><math><mi>y</mi><mo>∈</mo><mi>X</mi></math></span>. By construction, <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> has <span><math><mi>N</mi><mo>+</mo><mn>1</mn></math></span> eigenspaces. By construction, <em>A</em> acts on these eigenspaces in a (block) tridiagonal fashion. We show that <em>A</em> is diagonalizable, with <span><math><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn></math></span> eigenspaces. We show that <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> acts on these eigenspaces in a (block) tridiagonal fashion. Using this action, we show that <em>A</em> is <em>Q</em>-polynomial. We show that <span><math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> satisfy a pair of relations called the tridiagonal relations. We consider the subalgebra <em>T</em> of
本文的目的是展示衰减空间正集 Aq(N,M) 的 Q 多项式结构。正集 Aq(N,M) 简述如下。从有限域上具有 q 个元素的 (N+M) 维向量空间 H 开始。Aq(N,M)的顶点集 X 由与 h 有零交集的 H 子空间组成。Q 多项式结构涉及两个矩阵 A,A⁎∈MatX(C),其条目如下。对于 y,z∈X,矩阵 A 有 (y,z) 项 1(如果 y 覆盖了 z);qdimy(如果 z 覆盖了 y);0(如果 y,z 都没有覆盖另一个)。矩阵 A⁎ 是对角线,对于所有 y∈X 都有 (y,y) 项 q-dimy。根据构造,A⁎ 有 N+1 个特征空间。根据构造,A 以(分块)三对角方式作用于这些特征空间。我们证明 A 是可对角的,有 2N+1 个特征空间。我们证明 A⁎ 以(块)三对角方式作用于这些特征空间。利用这一作用,我们证明 A 是 Q 多项式。我们证明 A、A⁎ 满足一对称为三对角关系的关系。我们考虑由 A,A⁎ 生成的 MatX(C) 子代数 T。我们证明,A,A⁎ 作为伦纳德对作用于每个不可还原 T 模块。
{"title":"A Q-polynomial structure for the Attenuated Space poset Aq(N,M)","authors":"Paul Terwilliger","doi":"10.1016/j.jcta.2024.105872","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105872","url":null,"abstract":"<div><p>The goal of this article is to display a <em>Q</em>-polynomial structure for the Attenuated Space poset <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span>. The poset <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span> is briefly described as follows. Start with an <span><math><mo>(</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo>)</mo></math></span>-dimensional vector space <em>H</em> over a finite field with <em>q</em> elements. Fix an <em>M</em>-dimensional subspace <em>h</em> of <em>H</em>. The vertex set <em>X</em> of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span> consists of the subspaces of <em>H</em> that have zero intersection with <em>h</em>. The partial order on <em>X</em> is the inclusion relation. The <em>Q</em>-polynomial structure involves two matrices <span><math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>∈</mo><msub><mrow><mi>Mat</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> with the following entries. For <span><math><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>X</mi></math></span> the matrix <em>A</em> has <span><math><mo>(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></math></span>-entry 1 (if <em>y</em> covers <em>z</em>); <span><math><msup><mrow><mi>q</mi></mrow><mrow><mrow><mi>dim</mi></mrow><mspace></mspace><mi>y</mi></mrow></msup></math></span> (if <em>z</em> covers <em>y</em>); and 0 (if neither of <span><math><mi>y</mi><mo>,</mo><mi>z</mi></math></span> covers the other). The matrix <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is diagonal, with <span><math><mo>(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span>-entry <span><math><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mrow><mi>dim</mi></mrow><mspace></mspace><mi>y</mi></mrow></msup></math></span> for all <span><math><mi>y</mi><mo>∈</mo><mi>X</mi></math></span>. By construction, <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> has <span><math><mi>N</mi><mo>+</mo><mn>1</mn></math></span> eigenspaces. By construction, <em>A</em> acts on these eigenspaces in a (block) tridiagonal fashion. We show that <em>A</em> is diagonalizable, with <span><math><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn></math></span> eigenspaces. We show that <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> acts on these eigenspaces in a (block) tridiagonal fashion. Using this action, we show that <em>A</em> is <em>Q</em>-polynomial. We show that <span><math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> satisfy a pair of relations called the tridiagonal relations. We consider the subalgebra <em>T</em> of ","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105872"},"PeriodicalIF":1.1,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139713906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A graph is determined by its spectrum if there is not another graph with the same spectrum. Cámara and Haemers proved that the graph , obtained from the complete graph with n vertices by deleting all edges of a cycle with k vertices, is determined by its spectrum for , but not for . In this paper, we show that is the unique exception for the spectral determination of .
如果不存在另一个具有相同频谱的图,则该图由其频谱决定。Cámara 和 Haemers 证明,通过删除具有 k 个顶点的循环 Ck 的所有边,从具有 n 个顶点的完整图 Kn 得到的图 Kn∖Ck,在 k∈{3,4,5} 时由其谱决定,但在 k=6 时则不是。在本文中,我们将证明 k=6 是 Kn∖Ck 的谱确定性的唯一例外。
{"title":"Spectral characterization of the complete graph removing a cycle","authors":"Muhuo Liu , Xiaofeng Gu , Haiying Shan , Zoran Stanić","doi":"10.1016/j.jcta.2024.105868","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105868","url":null,"abstract":"<div><p>A graph is determined by its spectrum if there is not another graph with the same spectrum. Cámara and Haemers proved that the graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, obtained from the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <em>n</em> vertices by deleting all edges of a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> with <em>k</em> vertices, is determined by its spectrum for <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></math></span>, but not for <span><math><mi>k</mi><mo>=</mo><mn>6</mn></math></span>. In this paper, we show that <span><math><mi>k</mi><mo>=</mo><mn>6</mn></math></span> is the unique exception for the spectral determination of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105868"},"PeriodicalIF":1.1,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139713905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete polymatroid, its toric ring is studied deeply for several classes of polymatroids.
{"title":"The divisor class group of a discrete polymatroid","authors":"Jürgen Herzog , Takayuki Hibi , Somayeh Moradi , Ayesha Asloob Qureshi","doi":"10.1016/j.jcta.2024.105869","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105869","url":null,"abstract":"<div><p>In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete polymatroid, its toric ring is studied deeply for several classes of polymatroids.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105869"},"PeriodicalIF":1.1,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139709553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-02DOI: 10.1016/j.jcta.2024.105865
Vsevolod F. Lev
It is well-known that for a prime and integer , the maximum possible size of a sum-free subset of the elementary abelian group is . However, the matching stability result is known for only. We consider the first open case showing that if is a sum-free subset with , then there are a subgroup of size and an element such that .
{"title":"Large sum-free sets in Z5n","authors":"Vsevolod F. Lev","doi":"10.1016/j.jcta.2024.105865","DOIUrl":"10.1016/j.jcta.2024.105865","url":null,"abstract":"<div><p>It is well-known that for a prime <span><math><mi>p</mi><mo>≡</mo><mn>2</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span> and integer <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span><span>, the maximum possible size of a sum-free subset of the elementary abelian group </span><span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mspace></mspace><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>. However, the matching stability result is known for <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> only. We consider the first open case <span><math><mi>p</mi><mo>=</mo><mn>5</mn></math></span> showing that if <span><math><mi>A</mi><mo>⊆</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>5</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is a sum-free subset with <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>></mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⋅</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, then there are a subgroup <span><math><mi>H</mi><mo><</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>5</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> of size <span><math><mo>|</mo><mi>H</mi><mo>|</mo><mo>=</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> and an element <span><math><mi>e</mi><mo>∉</mo><mi>H</mi></math></span> such that <span><math><mi>A</mi><mo>⊆</mo><mo>(</mo><mi>e</mi><mo>+</mo><mi>H</mi><mo>)</mo><mo>∪</mo><mo>(</mo><mo>−</mo><mi>e</mi><mo>+</mo><mi>H</mi><mo>)</mo></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105865"},"PeriodicalIF":1.1,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139660492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-01DOI: 10.1016/j.jcta.2024.105866
Carmen Amarra , Alice Devillers , Cheryl E. Praeger
More than 30 years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive 2-design, with blocks of size k, could leave invariant a nontrivial point-partition, but only if the number of points was bounded in terms of k. Since then examples have been found where there are two nontrivial point partitions, either forming a chain of partitions, or forming a grid structure on the point set. We show, by construction of infinite families of designs, that there is no limit on the length of a chain of invariant point partitions for a block-transitive 2-design. We introduce the notion of an ‘array’ of a set of points which describes how the set interacts with parts of the various partitions, and we obtain necessary and sufficient conditions in terms of the ‘array’ of a point set, relative to a partition chain, for it to be a block of such a design.
30 多年前,德兰切尔和多延证明,大小为 k 块的块变换 2 设计的自动形群可以使一个非难点分区保持不变,但前提是点的数量以 k 为界。自那时起,人们发现了有两个非难点分区的例子,它们要么形成了分区链,要么在点集中形成了网格结构。我们通过构建无穷设计族证明,对于块过渡 2 设计,不变点分区链的长度没有限制。我们引入了点集 "阵列 "的概念,它描述了点集如何与不同分区的部分相互作用,我们还获得了点集 "阵列 "相对于分区链的必要条件和充分条件,从而使其成为这种设计的区块。
{"title":"Block-transitive 2-designs with a chain of imprimitive point-partitions","authors":"Carmen Amarra , Alice Devillers , Cheryl E. Praeger","doi":"10.1016/j.jcta.2024.105866","DOIUrl":"10.1016/j.jcta.2024.105866","url":null,"abstract":"<div><p>More than 30 years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive 2-design, with blocks of size <em>k</em>, could leave invariant a nontrivial point-partition, but only if the number of points was bounded in terms of <em>k</em>. Since then examples have been found where there are two nontrivial point partitions, either forming a chain of partitions, or forming a grid structure on the point set. We show, by construction of infinite families of designs, that there is no limit on the length of a chain of invariant point partitions for a block-transitive 2-design. We introduce the notion of an ‘array’ of a set of points which describes how the set interacts with parts of the various partitions, and we obtain necessary and sufficient conditions in terms of the ‘array’ of a point set, relative to a partition chain, for it to be a block of such a design.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105866"},"PeriodicalIF":1.1,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000050/pdfft?md5=4350b32c706696ea2c3f719fb15cd8bc&pid=1-s2.0-S0097316524000050-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139660473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-01DOI: 10.1016/j.jcta.2024.105867
Lyuben Lichev , Sammy Luo
Gyárfás famously showed that in every r-coloring of the edges of the complete graph , there is a monochromatic connected component with at least vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for k-uniform hypergraphs. Over a wide range of extensions of the definition of connectivity to higher uniformities, we provide both upper and lower bounds for the size of the largest monochromatic component that are tight up to a factor of as the number of colors grows. We further generalize these questions to ask about counts of vertex s-sets contained within the edges of large monochromatic components. We conclude with more precise results in the particular case of two colors.
Gyárfás 的著名研究表明,在完整图 Kn 的边的每 r 种着色中,都存在一个至少有 nr-1 个顶点的单色连通部分。最近,Conlon、Tyomkyn 和第二位作者的一项研究解决了具有许多边的单色连通成分的类似问题。在本文中,我们研究了这些问题在 k-uniform 超图中的推广。在将连通性定义扩展到更高均匀性的广泛范围内,我们为最大单色成分的大小提供了上界和下界,随着颜色数量的增加,上界和下界的紧密程度可达 1+o(1)。我们进一步将这些问题推广到大型单色分量边缘中包含的顶点 s 集的计数。最后,我们将针对两种颜色的特殊情况给出更精确的结果。
{"title":"Large monochromatic components in colorings of complete hypergraphs","authors":"Lyuben Lichev , Sammy Luo","doi":"10.1016/j.jcta.2024.105867","DOIUrl":"10.1016/j.jcta.2024.105867","url":null,"abstract":"<div><p>Gyárfás famously showed that in every <em>r</em>-coloring of the edges of the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, there is a monochromatic connected component with at least <span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span> vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for <em>k</em><span>-uniform hypergraphs<span>. Over a wide range of extensions of the definition of connectivity to higher uniformities, we provide both upper and lower bounds for the size of the largest monochromatic component that are tight up to a factor of </span></span><span><math><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> as the number of colors grows. We further generalize these questions to ask about counts of vertex <em>s</em>-sets contained within the edges of large monochromatic components. We conclude with more precise results in the particular case of two colors.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105867"},"PeriodicalIF":1.1,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139660447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}