{"title":"Classical groups as flag-transitive automorphism groups of 2-designs with λ = 2","authors":"Seyed Hassan Alavi , Mohsen Bayat , Ashraf Daneshkhah , Marjan Tadbirinia","doi":"10.1016/j.jcta.2024.105892","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105892","url":null,"abstract":"<div><p>In this article, we study 2-designs with <span><math><mi>λ</mi><mo>=</mo><mn>2</mn></math></span> admitting a flag-transitive and point-primitive almost simple automorphism group <em>G</em> with socle <em>X</em> a finite simple classical group of Lie type. We prove that such a design belongs to an infinite family of 2-designs with parameter set <span><math><mo>(</mo><mo>(</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> and <span><math><mi>X</mi><mo>=</mo><msub><mrow><mi>PSL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mn>3</mn><mo>)</mo></math></span> for some <span><math><mi>n</mi><mo>⩾</mo><mn>3</mn></math></span>, or <span><math><mi>X</mi><mo>=</mo><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> with point-stabiliser <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mo>(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mi>gcd</mi><mo></mo><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msub></math></span>, or it is isomorphic to the 2-design with parameter set <span><math><mo>(</mo><mn>6</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>7</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>10</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>11</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>28</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>28</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>36</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> or <span><math><mo>(</mo><mn>126</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105892"},"PeriodicalIF":1.1,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140546056","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":"Further q-reflections on the modulo 9 Kanade–Russell (conjectural) identities","authors":"Stepan Konenkov","doi":"10.1016/j.jcta.2024.105894","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105894","url":null,"abstract":"<div><p>We examine four identities conjectured by Dean Hickerson which complement five modulo 9 Kanade–Russell identities, and we build up a profile of new identities and new conjectures similar to those found by Ali Uncu and Wadim Zudilin.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105894"},"PeriodicalIF":1.1,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140540286","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":"A geometric proof for the root-independence of the greedoid polynomial of Eulerian branching greedoids","authors":"Lilla Tóthmérész","doi":"10.1016/j.jcta.2024.105891","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105891","url":null,"abstract":"<div><p>We define the root polytope of a regular oriented matroid, and show that the greedoid polynomial of an Eulerian branching greedoid rooted at vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is equivalent to the <span><math><msup><mrow><mi>h</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-polynomial of the root polytope of the dual of the graphic matroid.</p><p>As the definition of the root polytope is independent of the vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, this gives a geometric proof for the root-independence of the greedoid polynomial for Eulerian branching greedoids, a fact which was first proved by Swee Hong Chan, Kévin Perrot and Trung Van Pham using sandpile models. We also obtain that the greedoid polynomial does not change if we reverse every edge of an Eulerian digraph.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105891"},"PeriodicalIF":1.1,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S009731652400030X/pdfft?md5=24b6ae0df2f0ec32e3f3bfc6f52f2e2c&pid=1-s2.0-S009731652400030X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140347134","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}
Chenya Zhao , Binwei Zhao , Yanxun Chang , Tao Feng , Xiaomiao Wang , Menglong Zhang
{"title":"Cyclic relative difference families with block size four and their applications","authors":"Chenya Zhao , Binwei Zhao , Yanxun Chang , Tao Feng , Xiaomiao Wang , Menglong Zhang","doi":"10.1016/j.jcta.2024.105890","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105890","url":null,"abstract":"<div><p>Given a subgroup <em>H</em> of a group <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mo>+</mo><mo>)</mo></math></span>, a <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span> difference family (DF) is a set <span><math><mi>F</mi></math></span> of <em>k</em>-subsets of <em>G</em> such that <span><math><mo>{</mo><mi>f</mi><mo>−</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>:</mo><mi>f</mi><mo>,</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>F</mi><mo>,</mo><mi>f</mi><mo>≠</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><mi>F</mi><mo>∈</mo><mi>F</mi><mo>}</mo><mo>=</mo><mi>G</mi><mo>∖</mo><mi>H</mi></math></span>. Let <span><math><mi>g</mi><msub><mrow><mi>Z</mi></mrow><mrow><mi>g</mi><mi>h</mi></mrow></msub></math></span> be the subgroup of order <em>h</em> in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>g</mi><mi>h</mi></mrow></msub></math></span> generated by <em>g</em>. A <span><math><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>g</mi><mi>h</mi></mrow></msub><mo>,</mo><mi>g</mi><msub><mrow><mi>Z</mi></mrow><mrow><mi>g</mi><mi>h</mi></mrow></msub><mo>,</mo><mi>k</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-DF is called cyclic and written as a <span><math><mo>(</mo><mi>g</mi><mi>h</mi><mo>,</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-CDF. This paper shows that for <span><math><mi>h</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>6</mn><mo>}</mo></math></span>, there exists a <span><math><mo>(</mo><mi>g</mi><mi>h</mi><mo>,</mo><mi>h</mi><mo>,</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-CDF if and only if <span><math><mi>g</mi><mi>h</mi><mo>≡</mo><mi>h</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>12</mn><mo>)</mo></math></span>, <span><math><mi>g</mi><mo>⩾</mo><mn>4</mn></math></span> and <span><math><mo>(</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo>)</mo><mo>∉</mo><mo>{</mo><mo>(</mo><mn>9</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>,</mo><mo>(</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>}</mo></math></span>. As a corollary, it is shown that a 1-rotational Steiner system S<span><math><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi>v</mi><mo>)</mo></math></span> exists if and only if <span><math><mi>v</mi><mo>≡</mo><mn>4</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>12</mn><mo>)</mo></math></span> and <span><math><mi>v</mi><mo>≠</mo><mn>28</mn></math></span>. This solves the long-standing open problem on the existence of a 1-rotational S<span><math><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi>v</mi><mo>)</mo></math></span>. As another corollary, we establish the existence of an optimal <span><math><mo>(</mo><mi>v</mi><mo>,</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-optical orthogonal code with <span><math><mo>⌊</mo><mo>(</mo><mi>v</mi><mo>−</","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105890"},"PeriodicalIF":1.1,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140342255","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 complex genera, symmetric functions and multiple zeta values","authors":"Ping Li","doi":"10.1016/j.jcta.2024.105893","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105893","url":null,"abstract":"<div><p>We examine the coefficients in front of Chern numbers for complex genera, and pay special attention to the <span><math><msup><mrow><mtext>Td</mtext></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span>-genus, the Γ-genus as well as the Todd genus. Some related geometric applications to hyper-Kähler and Calabi-Yau manifolds are discussed. Along this line and building on the work of Doubilet in 1970s, various Hoffman-type formulas for multiple-(star) zeta values and transition matrices among canonical bases of the ring of symmetric functions can be uniformly treated in a more general framework.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105893"},"PeriodicalIF":1.1,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140344380","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":"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}
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