Journal of Combinatorial Theory Series A最新文献

{"title":"On nontrivial cross-t-intersecting families","authors":"Dongang He ,&nbsp;Anshui Li ,&nbsp;Biao Wu ,&nbsp;Huajun Zhang","doi":"10.1016/j.jcta.2025.106095","DOIUrl":"10.1016/j.jcta.2025.106095","url":null,"abstract":"<div><div>Two families <span><math><mi>A</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> and <span><math><mi>B</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>ℓ</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> are called nontrivial cross-<em>t</em>-intersecting if <span><math><mo>|</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>|</mo><mo>≥</mo><mi>t</mi></math></span> for all <span><math><mi>A</mi><mo>∈</mo><mi>A</mi></math></span>, <span><math><mi>B</mi><mo>∈</mo><mi>B</mi></math></span> and <span><math><mo>|</mo><msub><mrow><mo>⋂</mo></mrow><mrow><mi>A</mi><mo>∈</mo><mi>A</mi><mo>∪</mo><mi>B</mi></mrow></msub><mi>A</mi><mo>|</mo><mo>&lt;</mo><mi>t</mi></math></span>. In this paper we will determine the upper bound of <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>|</mo><mi>B</mi><mo>|</mo></math></span> for nontrivial cross-<em>t</em>-intersecting families <span><math><mi>A</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> and <span><math><mi>B</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>ℓ</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> for positive integers <em>n</em>, <em>k</em>, <em>ℓ</em> and <em>t</em> such that <span><math><mi>n</mi><mo>≥</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>k</mi><mo>−</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>,</mo><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>ℓ</mi><mo>−</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>}</mo></math></span> and <span><math><mi>t</mi><mo>≥</mo><mn>3</mn></math></span>. The structures of the extremal families attaining the upper bound are also characterized. As a byproduct of the main result in this paper, one product version of Erdős–Ko–Rado Theorem for two families of cross-<em>t</em>-intersecting can be easily obtained which gives a confirmative answer to one conjecture by Tokushige.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106095"},"PeriodicalIF":1.2,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144748719","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}

{"title":"General Theta function identities","authors":"Sun Kim","doi":"10.1016/j.jcta.2025.106094","DOIUrl":"10.1016/j.jcta.2025.106094","url":null,"abstract":"<div><div>Ramanujan's modular equations are closely associated with partition identities. In particular, the modular equations of prime degrees <span><math><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>11</mn></math></span>, 23 and the corresponding partition identities are of very elegant forms. These five modular equations were extensively generalized by Warnaar and the present author in the form of general theta function identities. In this paper, we provide further general theta function identities and present many partition identities as special cases.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106094"},"PeriodicalIF":0.9,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144679155","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}

{"title":"Stirling permutation codes. II","authors":"Shi-Mei Ma ,&nbsp;Hao Qi ,&nbsp;Jean Yeh ,&nbsp;Yeong-Nan Yeh","doi":"10.1016/j.jcta.2025.106093","DOIUrl":"10.1016/j.jcta.2025.106093","url":null,"abstract":"<div><div>In the context of Stirling polynomials, Gessel and Stanley introduced Stirling permutations, which have attracted extensive attention over the past decades. Recently, we introduced Stirling permutation codes and provided numerous equidistribution results as applications. The purpose of the present work is to further analyze Stirling permutation codes. First, we derive an expansion formula expressing the joint distribution of the types <em>A</em> and <em>B</em> descent statistics over the hyperoctahedral group, and we also find an interlacing property involving the zeros of its coefficient polynomials. Next, we prove a strong connection between signed permutations in the hyperoctahedral group and Stirling permutations. We also study unified generalizations of the trivariate second-order Eulerian and ascent-plateau polynomials. Using Stirling permutation codes, we provide expansion formulas for eight-variable and seventeen-variable polynomials, which imply several <em>e</em>-positive expansions and clarify the connection among several statistics. Our results generalize the results of Bóna, Chen-Fu, Dumont, Haglund-Visontai, Janson and Petersen.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106093"},"PeriodicalIF":0.9,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596379","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}

{"title":"Combinatorics on bi-γ-positivity of 1/k-Eulerian polynomials","authors":"Sherry H.F. Yan ,&nbsp;Xubo Yang ,&nbsp;Zhicong Lin","doi":"10.1016/j.jcta.2025.106092","DOIUrl":"10.1016/j.jcta.2025.106092","url":null,"abstract":"<div><div>The <span><math><mn>1</mn><mo>/</mo><mi>k</mi></math></span>-Eulerian polynomials <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo></math></span> were introduced as ascent polynomials over <em>k</em>-inversion sequences by Savage and Viswanathan. The bi-<em>γ</em>-positivity of the <span><math><mn>1</mn><mo>/</mo><mi>k</mi></math></span>-Eulerian polynomials <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo></math></span> was known but to give a combinatorial interpretation of the corresponding bi-<em>γ</em>-coefficients still remains open. The study of the theme of bi-<em>γ</em>-positivity from a purely combinatorial aspect was proposed by Athanasiadis. In this paper, we provide a combinatorial interpretation for the bi-<em>γ</em>-coefficients of <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo></math></span> by using the model of certain ordered labeled forests. Our combinatorial approach consists of three main steps:<ul><li><span>•</span><span><div>construct a bijection between <em>k</em>-Stirling permutations and certain forests that are named increasing pruned even <em>k</em>-ary forests;</div></span></li><li><span>•</span><span><div>introduce a generalized Foata–Strehl action on increasing pruned even <em>k</em>-ary trees which implies the longest ascent-plateau polynomials over <em>k</em>-Stirling permutations with initial letter 1 are <em>γ</em>-positive, a result that may have independent interest;</div></span></li><li><span>•</span><span><div>develop two crucial transformations on increasing pruned even <em>k</em>-ary forests to conclude our combinatorial interpretation.</div></span></li></ul></div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106092"},"PeriodicalIF":0.9,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144522733","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}

{"title":"Harmonic higher and extended weight enumerators","authors":"Thomas Britz ,&nbsp;Himadri Shekhar Chakraborty ,&nbsp;Tsuyoshi Miezaki","doi":"10.1016/j.jcta.2025.106090","DOIUrl":"10.1016/j.jcta.2025.106090","url":null,"abstract":"<div><div>In this paper, we present the harmonic generalizations of well-known polynomials of codes over finite fields, namely the higher weight enumerators and the extended weight enumerators, and we derive the correspondences between these weight enumerators. Moreover, we present the harmonic generalization of Greene's Theorem for the higher (resp. extended) weight enumerators. As an application of this Greene's-type theorem, we provide the MacWilliams-type identity for harmonic higher weight enumerators of codes over finite fields. Finally, we use this new identity to give a new proof of the Assmus-Mattson Theorem for subcode supports of linear codes over finite fields using harmonic higher weight enumerators.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106090"},"PeriodicalIF":0.9,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144489510","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}

{"title":"Proof of Lew's conjecture on the spectral gaps of simplicial complexes","authors":"Xiongfeng Zhan,&nbsp;Xueyi Huang,&nbsp;Huiqiu Lin","doi":"10.1016/j.jcta.2025.106091","DOIUrl":"10.1016/j.jcta.2025.106091","url":null,"abstract":"<div><div>As a generalization of graph Laplacians to higher dimensions, the combinatorial Laplacians of simplicial complexes have garnered increasing attention. Let <em>X</em> be a simplicial complex on <em>n</em> vertices, and let <span><math><mi>X</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> denote the set of all <em>k</em>-dimensional simplices of <em>X</em>. The <em>k</em>-th spectral gap <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is the smallest eigenvalue of the reduced <em>k</em>-dimensional Laplacian of <em>X</em>. For any <span><math><mi>k</mi><mo>≥</mo><mo>−</mo><mn>1</mn></math></span>, Lew (2020) <span><span>[24]</span></span> established a lower bound for <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>:<span><span><span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>≥</mo><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo><mrow><mo>(</mo><munder><mi>min</mi><mrow><mi>σ</mi><mo>∈</mo><mi>X</mi><mo>(</mo><mi>k</mi><mo>)</mo></mrow></munder><mo>⁡</mo><msub><mrow><mi>deg</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>⁡</mo><mo>(</mo><mi>σ</mi><mo>)</mo><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>−</mo><mi>d</mi><mi>n</mi><mo>≥</mo><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>−</mo><mi>d</mi><mi>n</mi><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>deg</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>⁡</mo><mo>(</mo><mi>σ</mi><mo>)</mo></math></span> and <em>d</em> denote the degree of <em>σ</em> in <em>X</em> and the maximal dimension of a missing face of <em>X</em>, respectively. In this paper, we identify the unique simplicial complex that achieves the lower bound of the <em>k</em>-th spectral gap, <span><math><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>−</mo><mi>d</mi><mi>n</mi></math></span>, for some <em>k</em>, thereby confirming a conjecture proposed by Lew.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106091"},"PeriodicalIF":0.9,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144306865","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}

{"title":"Avoiding short progressions in Euclidean Ramsey theory","authors":"Gabriel Currier ,&nbsp;Kenneth Moore ,&nbsp;Chi Hoi Yip","doi":"10.1016/j.jcta.2025.106080","DOIUrl":"10.1016/j.jcta.2025.106080","url":null,"abstract":"<div><div>We provide a general framework to construct colorings avoiding short monochromatic arithmetic progressions in Euclidean Ramsey theory. Specifically, if <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> denotes <em>m</em> collinear points with consecutive points of distance one apart, we say that <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>↛</mo><mo>(</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></math></span> if there is a red/blue coloring of <em>n</em>-dimensional Euclidean space that avoids red congruent copies of <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> and blue congruent copies of <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>. We show that <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>↛</mo><mo>(</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>20</mn></mrow></msub><mo>)</mo></math></span>, improving the best-known result <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>↛</mo><mo>(</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1177</mn></mrow></msub><mo>)</mo></math></span> by Führer and Tóth, and also establish <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>↛</mo><mo>(</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>14</mn></mrow></msub><mo>)</mo></math></span> and <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>↛</mo><mo>(</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>)</mo></math></span> in the spirit of the classical result <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>↛</mo><mo>(</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>)</mo></math></span> due to Erdős et al. We also show a number of similar 3-coloring results, as well as <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>↛</mo><mo>(</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mi>α</mi><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>6889</mn></mrow></msub><mo>)</mo></math></span>, where <em>α</em> is an arbitrary positive real number. This final result answers a question of Führer and Tóth in the positive.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106080"},"PeriodicalIF":0.9,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144262859","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}

{"title":"Perfect codes of bi-Cayley graphs","authors":"Yan Wang,&nbsp;Kai Yuan,&nbsp;Jian-Xun Li","doi":"10.1016/j.jcta.2025.106079","DOIUrl":"10.1016/j.jcta.2025.106079","url":null,"abstract":"<div><div>A bi-Cayley graph is a graph that has a semiregular group <em>H</em> of automorphisms having exactly two orbits on vertices, and it is called an algebraically Cayley graph if its full automorphism group contains a regular subgroup <em>G</em> such that <em>H</em> is a subgroup of <em>G</em>. An independent vertex subset of a graph is called a perfect code if each vertex outside of this subset is adjacent to exactly one vertex in it. In this paper, we give a necessary and sufficient condition for a bi-Cayley graph to be an algebraically Cayley graph, and perfect codes of such bi-Cayley graphs can be determined by the theory of perfect codes in Cayley graphs. Equivalent conditions for subsets to be perfect codes of regular (in terms of graph theory) bi-Cayley graphs are also given.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106079"},"PeriodicalIF":0.9,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144254272","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}

{"title":"The Frankl-Pach upper bound is not tight for any uniformity","authors":"Gennian Ge ,&nbsp;Zixiang Xu ,&nbsp;Chi Hoi Yip ,&nbsp;Shengtong Zhang ,&nbsp;Xiaochen Zhao","doi":"10.1016/j.jcta.2025.106078","DOIUrl":"10.1016/j.jcta.2025.106078","url":null,"abstract":"<div><div>For any positive integers <span><math><mi>n</mi><mo>≥</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>≥</mo><mn>3</mn></math></span>, what is the maximum size of a <span><math><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-uniform set system in <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span> with VC-dimension at most <em>d</em>? In 1984, Frankl and Pach initiated the study of this fundamental problem and provided an upper bound <span><math><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>d</mi></mtd></mtr></mtable><mo>)</mo></math></span> via an elegant algebraic proof. Surprisingly, in 2007, Mubayi and Zhao showed that when <em>n</em> is sufficiently large and <em>d</em> is a prime power, the Frankl-Pach upper bound is not tight. They also remarked that their method requires <em>d</em> to be a prime power, and asked for new ideas to improve the Frankl-Pach upper bound without extra assumptions on <em>n</em> and <em>d</em>.</div><div>In this paper, we provide an improvement for any <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>d</mi><mo>+</mo><mn>2</mn></math></span>, which demonstrates that the long-standing Frankl-Pach upper bound <span><math><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>d</mi></mtd></mtr></mtable><mo>)</mo></math></span> is not tight for any uniformity. Our proof combines a simple yet powerful polynomial method and structural analysis.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106078"},"PeriodicalIF":0.9,"publicationDate":"2025-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144212706","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}

{"title":"Level of regions for deformed braid arrangements","authors":"Yanru Chen ,&nbsp;Houshan Fu ,&nbsp;Suijie Wang ,&nbsp;Jinxing Yang","doi":"10.1016/j.jcta.2025.106077","DOIUrl":"10.1016/j.jcta.2025.106077","url":null,"abstract":"&lt;div&gt;&lt;div&gt;This paper primarily investigates a specific type of deformation of the braid arrangement in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, denoted by &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be the number of regions of level &lt;em&gt;l&lt;/em&gt; in &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; the corresponding exponential generating function. Using the weighted digraph model introduced by Hetyei, we establish a bijection between regions of level &lt;em&gt;l&lt;/em&gt; in &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; and valid &lt;em&gt;m&lt;/em&gt;-acyclic weighted digraphs on the vertex set &lt;span&gt;&lt;math&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with exactly &lt;em&gt;l&lt;/em&gt; strong components. Based on this bijection, we obtain that the sequence &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is of binomial type. In addition, the values &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; provide a combinatorial interpretation for the coefficients when the characteristic polynomial of &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; is expanded in terms of &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. In particular, if &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; for non-negative integers &lt;em&gt;a&lt;/em&gt; and &lt;em&gt;b&lt;/em&gt; with &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, we show that the characteristic polynomial of &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; has a single real root 0 of multiplicity one when &lt;em&gt;n&lt;/em&gt; is odd, and has one more real root &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; of multiplicity one whe","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106077"},"PeriodicalIF":0.9,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144212707","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}