{"title":"Odd-sunflowers","authors":"Peter Frankl , János Pach , Dömötör Pálvölgyi","doi":"10.1016/j.jcta.2024.105889","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105889","url":null,"abstract":"<div><p>Extending the notion of sunflowers, we call a family of at least two sets an <em>odd-sunflower</em> if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erdős–Szemerédi conjecture, recently proved by Naslund and Sawin, that there is a constant <span><math><mi>μ</mi><mo><</mo><mn>2</mn></math></span> such that every family of subsets of an <em>n</em>-element set that contains no odd-sunflower consists of at most <span><math><msup><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> sets. We construct such families of size at least <span><math><msup><mrow><mn>1.5021</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>. We also characterize minimal odd-sunflowers of triples.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105889"},"PeriodicalIF":1.1,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000281/pdfft?md5=4524ddd068e6ba4b9569281736257e67&pid=1-s2.0-S0097316524000281-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140180931","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}
{"title":"Extremal Peisert-type graphs without the strict-EKR property","authors":"Sergey Goryainov , Chi Hoi Yip","doi":"10.1016/j.jcta.2024.105887","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105887","url":null,"abstract":"<div><p>It is known that Paley graphs of square order have the strict-EKR property, that is, all maximum cliques are canonical cliques. Peisert-type graphs are natural generalizations of Paley graphs and some of them also have the strict-EKR property. Given a prime power <span><math><mi>q</mi><mo>≥</mo><mn>3</mn></math></span>, we study Peisert-type graphs of order <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> without the strict-EKR property and with the minimum number of edges and we call such graphs extremal. We determine number of edges in extremal graphs for each value of <em>q</em>. If <em>q</em> is a square or a cube, we show the uniqueness of the extremal graph and classify all maximum cliques explicitly. Moreover, when <em>q</em> is a square, we prove that there is no Hilton-Milner type result for the extremal graph, and show the tightness of the weight-distribution bound for both non-principal eigenvalues of this graph.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105887"},"PeriodicalIF":1.1,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140162511","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":"Phylogenetic degrees for claw trees","authors":"Rodica Andreea Dinu , Martin Vodička","doi":"10.1016/j.jcta.2024.105886","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105886","url":null,"abstract":"<div><p>Group-based models appear in algebraic statistics as mathematical models coming from evolutionary biology, namely in the study of mutations of genomes. Motivated also by applications, we are interested in determining the algebraic degrees of the phylogenetic varieties coming from these models. These algebraic degrees are called <em>phylogenetic degrees</em>. In this paper, we compute the phylogenetic degree of the variety <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> with <span><math><mi>G</mi><mo>∈</mo><mo>{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>}</mo></math></span> and any <em>n</em>-claw tree. As these varieties are toric, computing their phylogenetic degree relies on computing the volume of their associated polytopes <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. We apply combinatorial methods and we give concrete formulas for these volumes.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105886"},"PeriodicalIF":1.1,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000256/pdfft?md5=ffe34f8c1bb09972f0075bfe4ea95627&pid=1-s2.0-S0097316524000256-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140135009","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}
{"title":"Erdős-Ko-Rado theorem for bounded multisets","authors":"Jiaqi Liao , Zequn Lv , Mengyu Cao , Mei Lu","doi":"10.1016/j.jcta.2024.105888","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105888","url":null,"abstract":"<div><p>Let <span><math><mi>k</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi></math></span> be positive integers with <span><math><mi>k</mi><mo>⩾</mo><mn>2</mn></math></span>. A <em>k</em>-multiset of <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mrow><mi>m</mi></mrow></msub></math></span> is a collection of <em>k</em> integers from the set <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> in which the integers can appear more than once but at most <em>m</em> times. A family of such <em>k</em>-multisets is called an intersecting family if every pair of <em>k</em>-multisets from the family have non-empty intersection. A finite sequence of real numbers <span><math><mo>(</mo><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>n</mi></mrow></msub><mo>)</mo></math></span> is said to be unimodal if there is some <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>, such 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>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>⩽</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⩾</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⩾</mo><mo>…</mo><mo>⩾</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Given <span><math><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi></math></span>, denote <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub></math></span> as the coefficient of <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> in the generating function <span><math><msup><mrow><mo>(</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><msup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>ℓ</mi></mrow></msup></math></span>, where <span><math><mn>1</mn><mo>⩽</mo><mi>ℓ</mi><mo>⩽</mo><mi>n</mi></math></span>. In this paper, we first show that the sequence of <span><math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>)</mo></math></span> is unimodal. Then we use this as a tool to prove that the intersecting family in which every <em>k</em>-multiset contains a fixed element attains the maximum cardinality for <span><math><mi>n</mi><mo>⩾</mo><mi>k</mi><mo>+</mo><mrow><mo>⌈</mo><","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105888"},"PeriodicalIF":1.1,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140135010","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":"Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture","authors":"Young-Hoon Kiem , Donggun Lee","doi":"10.1016/j.jcta.2024.105884","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105884","url":null,"abstract":"<div><p>We introduce generalized Hessenberg varieties and establish basic facts. We show that the Tymoczko action of the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> on the cohomology of Hessenberg varieties extends to generalized Hessenberg varieties and that natural morphisms among them preserve the action. By analyzing natural morphisms and birational maps among generalized Hessenberg varieties, we give an elementary proof of the Shareshian-Wachs conjecture. Moreover we present a natural generalization of the Shareshian-Wachs conjecture that involves generalized Hessenberg varieties and provide an elementary proof. As a byproduct, we propose a generalized Stanley-Stembridge conjecture for <em>weighted</em> graphs. Our investigation into the birational geometry of generalized Hessenberg varieties enables us to modify them into much simpler varieties like projective spaces or permutohedral varieties by explicit sequences of blowups or projective bundle maps. Using this, we provide two algorithms to compute the <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-representations on the cohomology of generalized Hessenberg varieties. As an application, we compute representations on the low degree cohomology of some Hessenberg varieties.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105884"},"PeriodicalIF":1.1,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140031329","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 second largest eigenvalue of normal Cayley graphs on symmetric groups generated by cycles","authors":"Yuxuan Li, Binzhou Xia, Sanming Zhou","doi":"10.1016/j.jcta.2024.105885","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105885","url":null,"abstract":"<div><p>We study the normal Cayley graphs <span><math><mrow><mi>Cay</mi></mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>I</mi><mo>)</mo><mo>)</mo></math></span> on the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, where <span><math><mi>I</mi><mo>⊆</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> and <span><math><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>I</mi><mo>)</mo></math></span> is the set of all cycles in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with length in <em>I</em>. We prove that the strictly second largest eigenvalue of <span><math><mrow><mi>Cay</mi></mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>I</mi><mo>)</mo><mo>)</mo></math></span> can only be achieved by at most four irreducible representations of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and we determine further the multiplicity of this eigenvalue in several special cases. As a corollary, in the case when <em>I</em> contains neither <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> nor <em>n</em> we know exactly when <span><math><mrow><mi>Cay</mi></mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>I</mi><mo>)</mo><mo>)</mo></math></span> has the Aldous property, namely the strictly second largest eigenvalue is attained by the standard representation of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and we obtain that <span><math><mrow><mi>Cay</mi></mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>I</mi><mo>)</mo><mo>)</mo></math></span> does not have the Aldous property whenever <span><math><mi>n</mi><mo>∈</mo><mi>I</mi></math></span>. As another corollary of our main results, we prove a recent conjecture on the second largest eigenvalue of <span><math><mrow><mi>Cay</mi></mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><mi>k</mi><mo>}</mo><mo>)</mo><mo>)</mo></math></span> where <span><math><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>2</mn></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105885"},"PeriodicalIF":1.1,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000244/pdfft?md5=f945a709cc7931a6640e76d02ea647ea&pid=1-s2.0-S0097316524000244-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140014637","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}
{"title":"A short combinatorial proof of dimension identities of Erickson and Hunziker","authors":"Nishu Kumari","doi":"10.1016/j.jcta.2024.105883","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105883","url":null,"abstract":"<div><p>In a recent paper (<span>arXiv:2301.09744</span><svg><path></path></svg>), Erickson and Hunziker consider partitions in which the arm–leg difference is an arbitrary constant <em>m</em>. In previous works, these partitions are called <span><math><mo>(</mo><mo>−</mo><mi>m</mi><mo>)</mo></math></span>-asymmetric partitions. Regarding these partitions and their conjugates as highest weights, they prove an identity yielding an infinite family of dimension equalities between <span><math><msub><mrow><mi>gl</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>gl</mi></mrow><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msub></math></span> modules. Their proof proceeds by the manipulations of the hook content formula. We give a simple combinatorial proof of their result.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105883"},"PeriodicalIF":1.1,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139998969","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":"On the deepest cycle of a random mapping","authors":"Ljuben Mutafchiev , Steven Finch","doi":"10.1016/j.jcta.2024.105875","DOIUrl":"10.1016/j.jcta.2024.105875","url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the set of all mappings <span><math><mi>T</mi><mo>:</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo><mo>→</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. The corresponding graph of <em>T</em> is a union of disjoint connected unicyclic components. We assume that each <span><math><mi>T</mi><mo>∈</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is chosen uniformly at random (i.e., with probability <span><math><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>n</mi></mrow></msup></math></span>). The cycle of <em>T</em> contained within its largest component is called the <em>deepest</em> one. For any <span><math><mi>T</mi><mo>∈</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, let <span><math><msub><mrow><mi>ν</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math></span> denote the length of this cycle. In this paper, we establish the convergence in distribution of <span><math><msub><mrow><mi>ν</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></math></span> and find the limits of its expectation and variance as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>. For <em>n</em> large enough, we also show that nearly 55% of all cyclic vertices of a random mapping <span><math><mi>T</mi><mo>∈</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> lie in its deepest cycle and that a vertex from the longest cycle of <em>T</em> does not belong to its largest component with approximate probability 0.075.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105875"},"PeriodicalIF":1.1,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139937806","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":"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}
{"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}
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