Pub Date : 2025-04-30DOI: 10.1016/j.jcta.2025.106060
Aleksei L. Perezhogin , Vladimir N. Potapov , Anna A. Taranenko , Sergey Yu. Vladimirov
A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to 1. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if d is even, then the permanent of a d-dimensional polystochastic matrix of order 4 is positive, and for odd d, we give a complete characterization of d-dimensional polystochastic matrices with zero permanent.
{"title":"Characterization of polystochastic matrices of order 4 with zero permanent","authors":"Aleksei L. Perezhogin , Vladimir N. Potapov , Anna A. Taranenko , Sergey Yu. Vladimirov","doi":"10.1016/j.jcta.2025.106060","DOIUrl":"10.1016/j.jcta.2025.106060","url":null,"abstract":"<div><div>A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to 1. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if <em>d</em> is even, then the permanent of a <em>d</em>-dimensional polystochastic matrix of order 4 is positive, and for odd <em>d</em>, we give a complete characterization of <em>d</em>-dimensional polystochastic matrices with zero permanent.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106060"},"PeriodicalIF":0.9,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143886650","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 : 2025-04-29DOI: 10.1016/j.jcta.2025.106064
Sho Kubota , Kiyoto Yoshino
We completely characterize circulant graphs with valency up to 4 that admit perfect state transfer. Those of valency 3 do not admit it. On the other hand, circulant graphs with valency 4 admit perfect state transfer only in two infinite families: one discovered by Zhan and another new family, while no others do. The main tools for deriving these results are symmetry of graphs and eigenvalues. We describe necessary conditions for perfect state transfer to occur based on symmetry of graphs, which mathematically refers to automorphisms of graphs. As for eigenvalues, if perfect state transfer occurs, then certain eigenvalues of the corresponding isotropic random walks must be the halves of algebraic integers. Taking this into account, we utilize known results on the rings of integers of cyclotomic fields.
{"title":"Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks","authors":"Sho Kubota , Kiyoto Yoshino","doi":"10.1016/j.jcta.2025.106064","DOIUrl":"10.1016/j.jcta.2025.106064","url":null,"abstract":"<div><div>We completely characterize circulant graphs with valency up to 4 that admit perfect state transfer. Those of valency 3 do not admit it. On the other hand, circulant graphs with valency 4 admit perfect state transfer only in two infinite families: one discovered by Zhan and another new family, while no others do. The main tools for deriving these results are symmetry of graphs and eigenvalues. We describe necessary conditions for perfect state transfer to occur based on symmetry of graphs, which mathematically refers to automorphisms of graphs. As for eigenvalues, if perfect state transfer occurs, then certain eigenvalues of the corresponding isotropic random walks must be the halves of algebraic integers. Taking this into account, we utilize known results on the rings of integers of cyclotomic fields.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106064"},"PeriodicalIF":0.9,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143882707","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 : 2025-04-29DOI: 10.1016/j.jcta.2025.106062
Yongjiang Wu, Lihua Feng, Yongtao Li
<div><div>Two families <span><math><mi>F</mi></math></span> and <span><math><mi>G</mi></math></span> are called cross-intersecting if for every <span><math><mi>F</mi><mo>∈</mo><mi>F</mi></math></span> and <span><math><mi>G</mi><mo>∈</mo><mi>G</mi></math></span>, the intersection <span><math><mi>F</mi><mo>∩</mo><mi>G</mi></math></span> is non-empty. For any positive integers <em>n</em> and <em>k</em>, let <span><math><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></math></span> denote the family of all <em>k</em>-element subsets of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. Let <span><math><mi>t</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>n</mi></math></span> be non-negative integers with <span><math><mi>k</mi><mo>≥</mo><mi>s</mi><mo>+</mo><mn>1</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi><mo>+</mo><mi>t</mi></math></span>. In 2016, Frankl proved that if <span><math><mi>F</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span> and <span><math><mi>G</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> are cross-intersecting families, and <span><math><mi>F</mi></math></span> is <span><math><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-intersecting and <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≥</mo><mn>1</mn></math></span>, then <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mi>t</mi></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>1</mn></math></span>. Furthermore, Frankl conjectured that under an additional condition <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>k</mi><mo>+</mo><mi>t</mi><mo>+</mo><mi>s</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>⊆</mo><mi>F</mi></math></span>, the following inequality holds:<span><span><span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi><mo>+</mo><mi>s</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><munderover><mo>∑</mo><mr
{"title":"Proof of Frankl's conjecture on cross-intersecting families","authors":"Yongjiang Wu, Lihua Feng, Yongtao Li","doi":"10.1016/j.jcta.2025.106062","DOIUrl":"10.1016/j.jcta.2025.106062","url":null,"abstract":"<div><div>Two families <span><math><mi>F</mi></math></span> and <span><math><mi>G</mi></math></span> are called cross-intersecting if for every <span><math><mi>F</mi><mo>∈</mo><mi>F</mi></math></span> and <span><math><mi>G</mi><mo>∈</mo><mi>G</mi></math></span>, the intersection <span><math><mi>F</mi><mo>∩</mo><mi>G</mi></math></span> is non-empty. For any positive integers <em>n</em> and <em>k</em>, let <span><math><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></math></span> denote the family of all <em>k</em>-element subsets of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. Let <span><math><mi>t</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>n</mi></math></span> be non-negative integers with <span><math><mi>k</mi><mo>≥</mo><mi>s</mi><mo>+</mo><mn>1</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi><mo>+</mo><mi>t</mi></math></span>. In 2016, Frankl proved that if <span><math><mi>F</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span> and <span><math><mi>G</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> are cross-intersecting families, and <span><math><mi>F</mi></math></span> is <span><math><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-intersecting and <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≥</mo><mn>1</mn></math></span>, then <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mi>t</mi></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>1</mn></math></span>. Furthermore, Frankl conjectured that under an additional condition <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>k</mi><mo>+</mo><mi>t</mi><mo>+</mo><mi>s</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>⊆</mo><mi>F</mi></math></span>, the following inequality holds:<span><span><span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi><mo>+</mo><mi>s</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><munderover><mo>∑</mo><mr","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106062"},"PeriodicalIF":0.9,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143882709","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 : 2025-04-16DOI: 10.1016/j.jcta.2025.106051
Yuefeng Yang , Akihiro Munemasa , Kaishun Wang , Wenying Zhu
We classify certain non-symmetric commutative association schemes. As an application, we determine all the weakly distance-regular circulants of one type of arcs by using Schur rings. We also give the classification of primitive weakly distance-regular circulants.
{"title":"Weakly distance-regular circulants, I","authors":"Yuefeng Yang , Akihiro Munemasa , Kaishun Wang , Wenying Zhu","doi":"10.1016/j.jcta.2025.106051","DOIUrl":"10.1016/j.jcta.2025.106051","url":null,"abstract":"<div><div>We classify certain non-symmetric commutative association schemes. As an application, we determine all the weakly distance-regular circulants of one type of arcs by using Schur rings. We also give the classification of primitive weakly distance-regular circulants.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106051"},"PeriodicalIF":0.9,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143834212","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 : 2025-04-01Epub Date: 2024-11-26DOI: 10.1016/j.jcta.2024.105981
Yang Huang, Yuejian Peng
<div><div>Two families <span><math><mi>A</mi></math></span> and <span><math><mi>B</mi></math></span> are cross-intersecting if <span><math><mi>A</mi><mo>∩</mo><mi>B</mi><mo>≠</mo><mo>∅</mo></math></span> for any <span><math><mi>A</mi><mo>∈</mo><mi>A</mi></math></span> and <span><math><mi>B</mi><mo>∈</mo><mi>B</mi></math></span>. We call <em>t</em> families <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> pairwise cross-intersecting families if <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> are cross-intersecting for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>t</mi></math></span>. Additionally, if <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≠</mo><mo>∅</mo></math></span> for each <span><math><mi>j</mi><mo>∈</mo><mo>[</mo><mi>t</mi><mo>]</mo></math></span>, then we say that <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> are non-empty pairwise cross-intersecting. Let <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow></math></span> be non-empty pairwise cross-intersecting families with <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span>, <span><math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, <span><math><mi>n</mi><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></m
如果对于任意的 A∈A 和 B∈B 来说，A∩B≠∅Sm_2205↩，则两个族 A 和 B 是相交的。如果 Ai 和 Aj 在 1≤i<j≤t 时交叉，我们称 t 个族为 A1,A2,...,At 成对交叉族。此外，如果对于每个 j∈[t] Aj≠∅，那么我们说 A1,A2,...At 是非空的成对相交族。设 A1⊆([n]k1),A2⊆([n]k2),...,At⊆([n]kt)为非空成对相交族，t≥2，k1≥k2≥⋯≥kt，n≥k1+k2，d1,d2,...,dt 为正数。本文给出了∑j=1tdj|Aj|的尖锐上界，并描述了达到上界的族 A1,A2,...At 的特征。我们的结果统一了 Frankl 和 Tokushige (1992) [5]、Shi、Frankl 和 Qian (2022) [15]、Huang、Peng 和 Wang [10] 以及 Zhang 和 Feng [16] 的结果。此外，我们的结果可以应用于对某些 n<k1+k2 的处理，而之前已知的所有结果都没有这样的应用。在证明过程中，我们应用了 Kruskal 和 Katona 的一个结果，使我们只考虑其元素是按词典顺序排列的第一个 |Ai| 元素的 Ai 族。我们用一个单变量函数 fi(R) 限定∑i=1tdi|Ai|，其中 R 是按词法顺序排列的 Ai 的最后一个元素，并验证了 -fi(R)具有比极值结果更强的单调性。我们认为，除了极值结果之外，本文中函数的单模态性本身也很有趣。
{"title":"Non-empty pairwise cross-intersecting families","authors":"Yang Huang, Yuejian Peng","doi":"10.1016/j.jcta.2024.105981","DOIUrl":"10.1016/j.jcta.2024.105981","url":null,"abstract":"<div><div>Two families <span><math><mi>A</mi></math></span> and <span><math><mi>B</mi></math></span> are cross-intersecting if <span><math><mi>A</mi><mo>∩</mo><mi>B</mi><mo>≠</mo><mo>∅</mo></math></span> for any <span><math><mi>A</mi><mo>∈</mo><mi>A</mi></math></span> and <span><math><mi>B</mi><mo>∈</mo><mi>B</mi></math></span>. We call <em>t</em> families <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> pairwise cross-intersecting families if <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> are cross-intersecting for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>t</mi></math></span>. Additionally, if <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≠</mo><mo>∅</mo></math></span> for each <span><math><mi>j</mi><mo>∈</mo><mo>[</mo><mi>t</mi><mo>]</mo></math></span>, then we say that <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> are non-empty pairwise cross-intersecting. Let <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow></math></span> be non-empty pairwise cross-intersecting families with <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span>, <span><math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, <span><math><mi>n</mi><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></m","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105981"},"PeriodicalIF":0.9,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720004","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 : 2025-04-01Epub Date: 2024-11-27DOI: 10.1016/j.jcta.2024.105980
Van Long Phuoc Pham , Keshav Goyal , Han Mao Kiah
Transmit a codeword , that belongs to an -deletion-correcting code of length n, over a t-deletion channel for some . Levenshtein (2001) [10], proposed the problem of determining , the minimum number of distinct channel outputs required to uniquely reconstruct . Prior to this work, is known only when . Here, we provide an asymptotically exact solution for all values of ℓ and t. Specifically, we show that . In the special instances: where , we show that ; and when and , we show that . We also provide a conjecture on the exact value of for all values of n, ℓ, and t.
{"title":"Sequence reconstruction problem for deletion channels: A complete asymptotic solution","authors":"Van Long Phuoc Pham , Keshav Goyal , Han Mao Kiah","doi":"10.1016/j.jcta.2024.105980","DOIUrl":"10.1016/j.jcta.2024.105980","url":null,"abstract":"<div><div>Transmit a codeword <figure><img></figure>, that belongs to an <span><math><mo>(</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-deletion-correcting code of length <em>n</em>, over a <em>t</em>-deletion channel for some <span><math><mn>1</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>t</mi><mo><</mo><mi>n</mi></math></span>. Levenshtein (2001) <span><span>[10]</span></span>, proposed the problem of determining <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span>, the minimum number of distinct channel outputs required to uniquely reconstruct <figure><img></figure>. Prior to this work, <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> is known only when <span><math><mi>ℓ</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></math></span>. Here, we provide an asymptotically exact solution for all values of <em>ℓ</em> and <em>t</em>. Specifically, we show that <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>ℓ</mi></mrow></mtd></mtr><mtr><mtd><mi>ℓ</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mi>ℓ</mi><mo>)</mo><mo>!</mo></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mi>t</mi><mo>−</mo><mi>ℓ</mi></mrow></msup><mo>−</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>t</mi><mo>−</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span>. In the special instances: where <span><math><mi>ℓ</mi><mo>=</mo><mi>t</mi></math></span>, we show that <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>ℓ</mi></mrow></mtd></mtr><mtr><mtd><mi>ℓ</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span>; and when <span><math><mi>ℓ</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><mi>t</mi><mo>=</mo><mn>4</mn></math></span>, we show that <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>≤</mo><mn>20</mn><mi>n</mi><mo>−</mo><mn>150</mn></math></span>. We also provide a conjecture on the exact value of <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> for all values of <em>n</em>, <em>ℓ</em>, and <em>t</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105980"},"PeriodicalIF":0.9,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720012","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 : 2025-04-01Epub Date: 2024-11-26DOI: 10.1016/j.jcta.2024.105982
Byungchan Kim , Eunmi Kim
Let be the set of partitions of n into distinct parts, and be the sum of reciprocals of the parts of the partition λ. We show that as , Moreover, for , the set of ordinary partitions of n, we show that as , To prove these asymptotic formulas in a uniform manner, we derive a general asymptotic formula using Wright's circle method.
设 Dn 是将 n 分割为不同部分的集合，srp(λ) 是分割 λ 的各部分的倒数之和。 我们证明，当 n→∞ 时，E[srp(λ):λ∈Dn]∼log(3n)4andVar[srp(λ):λ∈Dn]∼π224。此外，对于 n 的普通分区集合 Pn，我们证明当 n→∞ 时，E[srp(λ):λ∈Pn]∼πn6andVar[srp(λ):λ∈Pn]∼π215n。为了统一证明这些渐近公式，我们利用赖特圆法推导出一个一般渐近公式。
{"title":"Distributions of reciprocal sums of parts in integer partitions","authors":"Byungchan Kim , Eunmi Kim","doi":"10.1016/j.jcta.2024.105982","DOIUrl":"10.1016/j.jcta.2024.105982","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the set of partitions of <em>n</em> into distinct parts, and <span><math><mi>srp</mi><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> be the sum of reciprocals of the parts of the partition <em>λ</em>. We show that as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>,<span><span><span><math><mi>E</mi><mo>[</mo><mi>srp</mi><mo>(</mo><mi>λ</mi><mo>)</mo><mo>:</mo><mi>λ</mi><mo>∈</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo><mo>∼</mo><mfrac><mrow><mi>log</mi><mo></mo><mo>(</mo><mn>3</mn><mi>n</mi><mo>)</mo></mrow><mrow><mn>4</mn></mrow></mfrac><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>Var</mi><mo>[</mo><mi>srp</mi><mo>(</mo><mi>λ</mi><mo>)</mo><mo>:</mo><mi>λ</mi><mo>∈</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo><mo>∼</mo><mfrac><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>24</mn></mrow></mfrac><mo>.</mo></math></span></span></span> Moreover, for <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the set of ordinary partitions of <em>n</em>, we show that as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>,<span><span><span><math><mi>E</mi><mo>[</mo><mi>srp</mi><mo>(</mo><mi>λ</mi><mo>)</mo><mo>:</mo><mi>λ</mi><mo>∈</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo><mo>∼</mo><mi>π</mi><msqrt><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></msqrt><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>Var</mi><mo>[</mo><mi>srp</mi><mo>(</mo><mi>λ</mi><mo>)</mo><mo>:</mo><mi>λ</mi><mo>∈</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo><mo>∼</mo><mfrac><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>15</mn></mrow></mfrac><mi>n</mi><mo>.</mo></math></span></span></span> To prove these asymptotic formulas in a uniform manner, we derive a general asymptotic formula using Wright's circle method.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105982"},"PeriodicalIF":0.9,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720007","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 : 2025-04-01Epub Date: 2024-11-16DOI: 10.1016/j.jcta.2024.105978
Takahiro Matsushita , Shun Wakatsuki
We show that the dominance complex of a graph G coincides with the combinatorial Alexander dual of the neighborhood complex of the complement of G. Using this, we obtain a relation between the chromatic number of G and the homology group of . We also obtain several known results related to dominance complexes from well-known facts of neighborhood complexes. After that, we suggest a new method for computing the homology groups of the dominance complexes, using independence complexes of simple graphs. We show that several known computations of homology groups of dominance complexes can be reduced to known computations of independence complexes. Finally, we determine the homology group of by determining the homotopy types of the independence complex of .
我们证明了图 G 的支配复数 D(G) 与 G 的补集的邻域复数 N(G‾) 的组合亚历山大对偶重合，并由此得到了 G 的色度数 χ(G) 与 D(G) 的同调群之间的关系。我们还从邻接复数的著名事实中得到了几个与支配复数有关的已知结果。之后，我们提出了一种利用简单图的独立复数计算支配复数同调群的新方法。我们证明了支配复数同调群的几种已知计算方法可以简化为独立复数的已知计算方法。最后，我们通过确定 Pn×P3×P2 独立复数的同调类型来确定 D(Pn×P3) 的同调群。
{"title":"Dominance complexes, neighborhood complexes and combinatorial Alexander duals","authors":"Takahiro Matsushita , Shun Wakatsuki","doi":"10.1016/j.jcta.2024.105978","DOIUrl":"10.1016/j.jcta.2024.105978","url":null,"abstract":"<div><div>We show that the dominance complex <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> coincides with the combinatorial Alexander dual of the neighborhood complex <span><math><mi>N</mi><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover><mo>)</mo></math></span> of the complement of <em>G</em>. Using this, we obtain a relation between the chromatic number <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of <em>G</em> and the homology group of <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. We also obtain several known results related to dominance complexes from well-known facts of neighborhood complexes. After that, we suggest a new method for computing the homology groups of the dominance complexes, using independence complexes of simple graphs. We show that several known computations of homology groups of dominance complexes can be reduced to known computations of independence complexes. Finally, we determine the homology group of <span><math><mi>D</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span> by determining the homotopy types of the independence complex of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105978"},"PeriodicalIF":0.9,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652948","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}
Let Γ be a Cayley graph on a finite group G, and let be the normalizer of (the right regular representation of G) in the full automorphism group of Γ. We say that Γ is a normal Cayley graph on G if , and that Γ is a normal edge-transitive Cayley graph on G if acts transitively on the edge set of Γ. In 1999, Praeger proved that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency 3 is normal. As an extension of this, in this paper, we prove that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency p is normal for each prime p. This, however, is not true for composite valency. We give a method to construct connected normal edge-transitive but non-normal Cayley graphs of certain groups, and using this, we prove that if G is either for an odd prime , or for , then there exists a connected normal edge-transitive but non-normal 8-valent Cayley graph of G.
{"title":"Normal edge-transitive Cayley graphs on non-abelian simple groups","authors":"Xing Zhang, Yan-Quan Feng, Fu-Gang Yin, Jin-Xin Zhou","doi":"10.1016/j.jcta.2025.106050","DOIUrl":"10.1016/j.jcta.2025.106050","url":null,"abstract":"<div><div>Let <em>Γ</em> be a Cayley graph on a finite group <em>G</em>, and let <span><math><msub><mrow><mi>N</mi></mrow><mrow><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> be the normalizer of <span><math><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> (the right regular representation of <em>G</em>) in the full automorphism group <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> of <em>Γ</em>. We say that <em>Γ</em> is a normal Cayley graph on <em>G</em> if <span><math><msub><mrow><mi>N</mi></mrow><mrow><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>=</mo><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span>, and that <em>Γ</em> is a normal edge-transitive Cayley graph on <em>G</em> if <span><math><msub><mrow><mi>N</mi></mrow><mrow><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> acts transitively on the edge set of <em>Γ</em>. In 1999, Praeger proved that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency 3 is normal. As an extension of this, in this paper, we prove that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency <em>p</em> is normal for each prime <em>p</em>. This, however, is not true for composite valency. We give a method to construct connected normal edge-transitive but non-normal Cayley graphs of certain groups, and using this, we prove that if <em>G</em> is either <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> for an odd prime <span><math><mi>q</mi><mo>≥</mo><mn>5</mn></math></span>, or <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>, then there exists a connected normal edge-transitive but non-normal 8-valent Cayley graph of <em>G</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106050"},"PeriodicalIF":0.9,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739311","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 : 2025-03-18DOI: 10.1016/j.jcta.2025.106035
Heng Huat Chan , Song Heng Chan
In this article, we discuss finite versions of Euler's pentagonal number identity, the Rogers-Ramanujan identities and present new proofs of the finite versions of the Andrews-Gordon identity and the Bressoud identity. We also investigate the finite version of Garvan's generalizations of Dyson's rank and discover a new one-variable extension of the Andrews-Gordon identity.
{"title":"Finite versions of the Andrews–Gordon identity and Bressoud's identity","authors":"Heng Huat Chan , Song Heng Chan","doi":"10.1016/j.jcta.2025.106035","DOIUrl":"10.1016/j.jcta.2025.106035","url":null,"abstract":"<div><div>In this article, we discuss finite versions of Euler's pentagonal number identity, the Rogers-Ramanujan identities and present new proofs of the finite versions of the Andrews-Gordon identity and the Bressoud identity. We also investigate the finite version of Garvan's generalizations of Dyson's rank and discover a new one-variable extension of the Andrews-Gordon identity.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106035"},"PeriodicalIF":0.9,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643721","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}
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