Journal of Combinatorial Theory Series A最新文献

{"title":"The geometry of intersecting codes and applications to additive combinatorics and factorization theory","authors":"Martino Borello ,&nbsp;Wolfgang Schmid ,&nbsp;Martin Scotti","doi":"10.1016/j.jcta.2025.106023","DOIUrl":"10.1016/j.jcta.2025.106023","url":null,"abstract":"<div><div>Intersecting codes are linear codes where every two nonzero codewords have non-trivially intersecting support. In this article we expand on the theory of this family of codes, by showing that nondegenerate intersecting codes correspond to sets of points (with multiplicities) in a projective space that are not contained in two hyperplanes. This correspondence allows the use of geometric arguments to demonstrate properties and provide constructions of intersecting codes. We improve on existing bounds on their length and provide explicit constructions of short intersecting codes. Finally, generalizing a link between coding theory and the theory of the Davenport constant (a combinatorial invariant of finite abelian groups), we provide new asymptotic bounds on the weighted 2-wise Davenport constant. These bounds then yield results on factorizations in rings of algebraic integers and related structures.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106023"},"PeriodicalIF":0.9,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510238","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":"Separable elements and splittings in Weyl groups of type B","authors":"Ming Liu,&nbsp;Houyi Yu","doi":"10.1016/j.jcta.2025.106021","DOIUrl":"10.1016/j.jcta.2025.106021","url":null,"abstract":"<div><div>Separable elements in Weyl groups are generalizations of the well-known class of separable permutations in symmetric groups. Gaetz and Gao showed that for any pair <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> of subsets of the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the multiplication map <span><math><mi>X</mi><mo>×</mo><mi>Y</mi><mo>→</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is a splitting (i.e., a length-additive bijection) of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> if and only if <em>X</em> is the generalized quotient of <em>Y</em> and <em>Y</em> is a principal lower order ideal in the right weak order generated by a separable element. They conjectured this result can be extended to all finite Weyl groups. In this paper, we classify all separable and minimal non-separable signed permutations in terms of forbidden patterns and confirm the conjecture of Gaetz and Gao for Weyl groups of type <em>B</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106021"},"PeriodicalIF":0.9,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510239","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":"Multivariate P- and/or Q-polynomial association schemes","authors":"Eiichi Bannai ,&nbsp;Hirotake Kurihara ,&nbsp;Da Zhao ,&nbsp;Yan Zhu","doi":"10.1016/j.jcta.2025.106025","DOIUrl":"10.1016/j.jcta.2025.106025","url":null,"abstract":"<div><div>The classification problem of <em>P</em>- and <em>Q</em>-polynomial association schemes has been one of the central problems in algebraic combinatorics. Generalizing the concept of <em>P</em>- and <em>Q</em>-polynomial association schemes to multivariate cases, namely to consider higher rank <em>P</em>- and <em>Q</em>-polynomial association schemes, has been tried by some authors, but it seems that so far there were neither very well-established definitions nor results. Very recently, Bernard, Crampé, d'Andecy, Vinet, and Zaimi <span><span>[4]</span></span>, defined bivariate <em>P</em>-polynomial association schemes, as well as bivariate <em>Q</em>-polynomial association schemes. In this paper, we study these concepts and propose a new modified definition concerning a general monomial order, which is more general and more natural and also easy to handle. We prove that there are many interesting families of examples of multivariate <em>P</em>- and/or <em>Q</em>-polynomial association schemes.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106025"},"PeriodicalIF":0.9,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510697","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 bijection related to Bressoud's conjecture","authors":"Y.H. Chen,&nbsp;Thomas Y. He","doi":"10.1016/j.jcta.2025.106032","DOIUrl":"10.1016/j.jcta.2025.106032","url":null,"abstract":"<div><div>Bressoud introduced the partition function <span><math><mi>B</mi><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>;</mo><mi>η</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>;</mo><mi>n</mi><mo>)</mo></math></span>, which counts the number of partitions with certain difference conditions. Bressoud posed a conjecture on the generating function for the partition function <span><math><mi>B</mi><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>;</mo><mi>η</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>;</mo><mi>n</mi><mo>)</mo></math></span> in multi-summation form. In this article, we introduce a bijection related to Bressoud's conjecture. As an application, we give the proof of a companion to the Göllnitz-Gordon identities.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106032"},"PeriodicalIF":0.9,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510241","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 de Bruijn rings and families of almost perfect maps","authors":"Peer Stelldinger","doi":"10.1016/j.jcta.2025.106030","DOIUrl":"10.1016/j.jcta.2025.106030","url":null,"abstract":"<div><div>De Bruijn tori, or perfect maps, are two-dimensional periodic arrays of letters from a finite alphabet, where each possible pattern of shape <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> appears exactly once in a single period. While the existence of certain de Bruijn tori, such as square tori with odd <span><math><mi>m</mi><mo>=</mo><mi>n</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>}</mo></math></span> and even alphabet sizes, remains unresolved, sub-perfect maps are often sufficient in applications like positional coding. These maps capture a large number of patterns, with each appearing at most once. While previous methods for generating such sub-perfect maps cover only a fraction of the possible patterns, we present a construction method for generating almost perfect maps for arbitrary pattern shapes and arbitrary non-prime alphabet sizes, including the above mentioned square tori with odd <span><math><mi>m</mi><mo>=</mo><mi>n</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>}</mo></math></span> as long that the alphabet size is non-prime. This is achieved through the introduction of de Bruijn rings, a minimal-height sub-perfect map and a formalization of the concept of families of almost perfect maps. The generated sub-perfect maps are easily decodable which makes them perfectly suitable for positional coding applications.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106030"},"PeriodicalIF":0.9,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510240","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":"An imperceptible connection between the Clebsch–Gordan coefficients of Uq(sl2) and the Terwilliger algebras of Grassmann graphs","authors":"Hau-Wen Huang","doi":"10.1016/j.jcta.2025.106028","DOIUrl":"10.1016/j.jcta.2025.106028","url":null,"abstract":"&lt;div&gt;&lt;div&gt;The Clebsch–Gordan coefficients of &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&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;/math&gt;&lt;/span&gt; are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism ♮ from the universal Hahn algebra &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; into &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&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;mo&gt;⊗&lt;/mo&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&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;/math&gt;&lt;/span&gt;. Let Ω denote a finite set of size &lt;em&gt;D&lt;/em&gt; and &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; denote the power set of Ω. It is generally known that &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; supports a &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&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;/math&gt;&lt;/span&gt;-module. Let &lt;em&gt;k&lt;/em&gt; denote an integer with &lt;span&gt;&lt;math&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and fix a &lt;em&gt;k&lt;/em&gt;-element subset &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; of Ω. By identifying &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;∖&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; this induces a &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&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;mo&gt;⊗&lt;/mo&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&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;/math&gt;&lt;/span&gt;-module structure on &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; denoted by &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Pulling back via ♮ the &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&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;mo&gt;⊗&lt;/mo&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sl&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;/math&gt;&lt;/span&gt;-module &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; forms an &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-module. When &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106028"},"PeriodicalIF":0.9,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510465","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":"More on r-cross t-intersecting families for vector spaces","authors":"Tian Yao ,&nbsp;Dehai Liu ,&nbsp;Kaishun Wang","doi":"10.1016/j.jcta.2025.106031","DOIUrl":"10.1016/j.jcta.2025.106031","url":null,"abstract":"<div><div>Let <em>V</em> be a finite dimensional vector space over a finite field. Suppose that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, …, <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> are <em>r</em>-cross <em>t</em>-intersecting families of <em>k</em>-subspaces of <em>V</em>. In this paper, we determine the extremal structure when <span><math><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></math></span> is maximum under the condition that <span><math><mi>dim</mi><mo>⁡</mo><mo>(</mo><msub><mrow><mo>⋂</mo></mrow><mrow><mi>F</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mi>F</mi><mo>)</mo><mo>&lt;</mo><mi>t</mi></math></span> for each <em>i</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106031"},"PeriodicalIF":0.9,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510698","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":"Regular ovoids and Cameron-Liebler sets of generators in polar spaces","authors":"Maarten De Boeck ,&nbsp;Jozefien D'haeseleer ,&nbsp;Morgan Rodgers","doi":"10.1016/j.jcta.2025.106029","DOIUrl":"10.1016/j.jcta.2025.106029","url":null,"abstract":"<div><div>Cameron-Liebler sets of generators in polar spaces were introduced a few years ago as natural generalisations of the Cameron-Liebler sets of subspaces in projective spaces. In this article we present the first two constructions of non-trivial Cameron-Liebler sets of generators in polar spaces. Also regular <em>m</em>-ovoids of <em>k</em>-spaces are introduced as a generalization of <em>m</em>-ovoids of polar spaces. They are used in one of the aforementioned constructions of Cameron-Liebler sets.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106029"},"PeriodicalIF":0.9,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143480765","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":"Truncated forms of MacMahon's q-series","authors":"Mircea Merca","doi":"10.1016/j.jcta.2025.106020","DOIUrl":"10.1016/j.jcta.2025.106020","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In 1920, Percy Alexander MacMahon defined the partition generating functions&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munder&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&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;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&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;&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&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;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munder&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&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;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&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;&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&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;mn&gt;2&lt;/mn&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&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;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&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;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&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;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; which have since played an important rol","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106020"},"PeriodicalIF":0.9,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143480764","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":"Simple geometric mitosis","authors":"Valentina Kiritchenko","doi":"10.1016/j.jcta.2025.106022","DOIUrl":"10.1016/j.jcta.2025.106022","url":null,"abstract":"<div><div>We construct simple geometric operations on faces of the Cayley sum of two polytopes. These operations can be thought of as convex geometric counterparts of divided difference operators in Schubert calculus. We show that these operations give a uniform construction of Knutson–Miller mitosis in the type <em>A</em> and Fujita mitosis in the type <em>C</em> on Kogan faces of Gelfand–Zetlin polytopes.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106022"},"PeriodicalIF":0.9,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474227","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}