{"title":"Real toric manifolds associated with chordal nestohedra","authors":"Suyoung Choi, Younghan Yoon","doi":"10.1016/j.jcta.2025.106102","DOIUrl":"10.1016/j.jcta.2025.106102","url":null,"abstract":"<div><div>This paper investigates the rational Betti numbers of real toric manifolds associated with chordal nestohedra. We consider the poset topology of a specific poset induced from a chordal building set, and show its EL-shellability. Based on this, we present an explicit description using alternating <span><math><mi>B</mi></math></span>-permutations for a chordal building set <span><math><mi>B</mi></math></span>, transforming the computing Betti numbers into a counting problem. This approach allows us to compute the <em>a</em>-number of a finite simple graph through permutation counting when the graph is chordal. In addition, we provide detailed computations for specific cases such as real Hochschild varieties corresponding to Hochschild polytopes.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106102"},"PeriodicalIF":1.2,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896676","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}
Yannik Eikmeier, Pamela Fleischmann, Mitja Kulczynski, Dirk Nowotka, Max Wiedenhöft
{"title":"Weighted prefix normal words","authors":"Yannik Eikmeier, Pamela Fleischmann, Mitja Kulczynski, Dirk Nowotka, Max Wiedenhöft","doi":"10.1016/j.jcta.2025.106101","DOIUrl":"10.1016/j.jcta.2025.106101","url":null,"abstract":"<div><div>A prefix normal word is a binary word whose prefixes contain at least as many 1s as any of its factors of the same length. Introduced by Fici and Lipták in 2011, the notion of prefix normality has been, thus far, only defined for words over the binary alphabet. In this work we investigate a generalisation for finite words over arbitrary finite alphabets, namely weighted prefix normality. We prove that weighted prefix normality is more expressive than binary prefix normality. Furthermore, we investigate the existence of a weighted prefix normal form, since weighted prefix normality comes with several new peculiarities that did not already occur in the binary case. We characterise these issues and finally present a standard technique to obtain a generalised prefix normal form for all words over arbitrary, finite alphabets. Additionally, we show a collection of results for the language of those prefix normal forms and extend the connection to Lyndon words and pre-necklaces to the general alphabet.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106101"},"PeriodicalIF":1.2,"publicationDate":"2025-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144860902","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":"Maximum Erdős-Ko-Rado sets of chambers and their antidesigns in vector-spaces of even dimension","authors":"Philipp Heering , Jesse Lansdown , Klaus Metsch","doi":"10.1016/j.jcta.2025.106098","DOIUrl":"10.1016/j.jcta.2025.106098","url":null,"abstract":"<div><div>A chamber of the vector space <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is a set <span><math><mo>{</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo></math></span> of subspaces of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> where <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊂</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊂</mo><mo>…</mo><mo>⊂</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> and <span><math><mi>dim</mi><mo></mo><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>i</mi></math></span> for <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>. By <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> we denote the graph whose vertices are the chambers of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> with two chambers <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mo>{</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mo>{</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo></math></span> adjacent in <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>, if <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msub><mo>=</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span> for <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>. The Erdős-Ko-Rado problem on chambers is equivalent to determining the structure of independent sets of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>. The independence number of this graph was determined in <span><span>[5]</span></span> for <em>n</em> even and given a subspace <em>P</em> of dimension one, the set of all chambers whose subspaces of dimension <span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> contain <em>P</em> attains the bound. The dual example of course also attains th","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106098"},"PeriodicalIF":1.2,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831267","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":"Intersection-union families","authors":"Peter Frankl , Jian Wang","doi":"10.1016/j.jcta.2025.106100","DOIUrl":"10.1016/j.jcta.2025.106100","url":null,"abstract":"<div><div>Let <span><math><msup><mrow><mn>2</mn></mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></msup></math></span> denote the power set of the <em>n</em>-set <span><math><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>. For positive integers <span><math><mi>n</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi></math></span>, <span><math><mi>n</mi><mo>≥</mo><mi>p</mi><mo>+</mo><mi>q</mi></math></span> let <span><math><mi>m</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> denote the maximum of <span><math><mo>|</mo><mi>F</mi><mo>|</mo></math></span> for a family <span><math><mi>F</mi><mo>⊂</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></msup></math></span> satisfying <span><math><mo>|</mo><mi>F</mi><mo>∩</mo><mi>G</mi><mo>|</mo><mo>≥</mo><mi>p</mi></math></span> and <span><math><mo>|</mo><mi>F</mi><mo>∪</mo><mi>G</mi><mo>|</mo><mo>≤</mo><mi>n</mi><mo>−</mo><mi>q</mi></math></span> for all <span><math><mi>F</mi><mo>,</mo><mi>G</mi><mo>∈</mo><mi>F</mi></math></span>. The exact value of <span><math><mi>m</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> has been known for half a century in the case <span><math><mi>p</mi><mo>=</mo><mn>1</mn></math></span> or <span><math><mi>q</mi><mo>=</mo><mn>1</mn></math></span>. Bang, Sharp and Winkler determined it in the case <span><math><mi>n</mi><mo>−</mo><mi>p</mi><mo>−</mo><mi>q</mi><mo>≤</mo><mn>3</mn></math></span>. The aim of the present paper is to establish the exact value of <span><math><mi>m</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><msup><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>p</mi><mo>−</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup></math></span> and also for <span><math><mi>n</mi><mo>−</mo><mi>p</mi><mo>−</mo><mi>q</mi><mo>=</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106100"},"PeriodicalIF":1.2,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831268","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 Weisfeiler-Leman stabilization of a tree","authors":"Jing Xu , Tatsuro Ito , Shuang-Dong Li","doi":"10.1016/j.jcta.2025.106099","DOIUrl":"10.1016/j.jcta.2025.106099","url":null,"abstract":"<div><div>For the Weisfeiler-Leman stabilization, we introduce a concept, which we call the coherent length, to measure how long it takes. We show that the coherent length is at most <em>8</em> for trees, using the structures of their <em>T</em>-algebras and of the centralizer algebras of their automorphism groups.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"218 ","pages":"Article 106099"},"PeriodicalIF":1.2,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144828429","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 some nonnormal Cayley graphs on symmetric groups","authors":"Yuxuan Li, Binzhou Xia, Sanming Zhou","doi":"10.1016/j.jcta.2025.106097","DOIUrl":"https://doi.org/10.1016/j.jcta.2025.106097","url":null,"abstract":"A Cayley graph on the symmetric group <mml:math altimg=\"si1.svg\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of <mml:math altimg=\"si1.svg\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>. For <mml:math altimg=\"si2.svg\"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"><</mml:mo><mml:mi>n</mml:mi></mml:math>, let <mml:math altimg=\"si267.svg\"><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> be the set of <ce:italic>k</ce:italic>-cycles of <mml:math altimg=\"si1.svg\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> moving every point in <mml:math altimg=\"si4.svg\"><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:math>. Recently, Siemons and Zalesski (2022) <ce:cross-ref ref>[26]</ce:cross-ref> posed a conjecture which is equivalent to saying that for any <mml:math altimg=\"si5.svg\"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>5</mml:mn></mml:math> and <mml:math altimg=\"si2.svg\"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"><</mml:mo><mml:mi>n</mml:mi></mml:math> the nonnormal Cayley graph <mml:math altimg=\"si6.svg\"><mml:mrow><mml:mi mathvariant=\"normal\">Cay</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math> on <mml:math altimg=\"si1.svg\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> with connection set <mml:math altimg=\"si267.svg\"><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> has the Aldous property. Solving this conjecture, we prove that all these graphs have the Aldous property except when (i) <mml:math altimg=\"si7.svg\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"22 1","pages":"106097"},"PeriodicalIF":1.1,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144900111","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}
Jia-Li Du , Yan-Quan Feng , Binzhou Xia , Da-Wei Yang
{"title":"The existence of m-Haar graphical representations","authors":"Jia-Li Du , Yan-Quan Feng , Binzhou Xia , Da-Wei Yang","doi":"10.1016/j.jcta.2025.106096","DOIUrl":"10.1016/j.jcta.2025.106096","url":null,"abstract":"<div><div>Extending the well-studied concept of graphical regular representations to bipartite graphs, a Haar graphical representation (HGR) of a group <em>G</em> is a bipartite graph whose automorphism group is isomorphic to <em>G</em> and acts semiregularly with the orbits giving the bipartition. The question of which groups admit an HGR was inspired by a closely related question of Estélyi and Pisanski in 2016, as well as Babai's work in 1980 on poset representations, and has been recently solved by Morris and Spiga. In this paper, we introduce the <em>m</em>-Haar graphical representation (<em>m</em>-HGR) as a natural generalization of HGR to <em>m</em>-partite graphs for <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span>, and explore the existence of <em>m</em>-HGRs for any fixed group. This inquiry represents a more robust version of the existence problem of G<em>m</em>SRs as addressed by Du, Feng and Spiga in 2020. Our main result is a complete classification of finite groups <em>G</em> without <em>m</em>-HGRs.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"218 ","pages":"Article 106096"},"PeriodicalIF":1.2,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144770884","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 nontrivial cross-t-intersecting families","authors":"Dongang He , Anshui Li , Biao Wu , 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><</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 , Hao Qi , Jean Yeh , 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}
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