Journal of Combinatorial Theory Series A最新文献

{"title":"On the number of subsequence sums related to the support of a sequence in finite abelian groups","authors":"Rui Wang,&nbsp;Han Chao,&nbsp;Jiangtao Peng","doi":"10.1016/j.jcta.2025.106124","DOIUrl":"10.1016/j.jcta.2025.106124","url":null,"abstract":"<div><div>Let <em>G</em> be a finite abelian group and <em>S</em> a sequence with elements of <em>G</em>. Let <span><math><mo>|</mo><mi>S</mi><mo>|</mo></math></span> denote the length of <em>S</em> and <span><math><mrow><mi>supp</mi></mrow><mo>(</mo><mi>S</mi><mo>)</mo></math></span> the set of all the distinct terms in <em>S</em>. For an integer <em>k</em> with <span><math><mi>k</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>]</mo></math></span>, let <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>⊂</mo><mi>G</mi></math></span> denote the set of group elements which can be expressed as a sum of a subsequence of <em>S</em> with length <em>k</em>. Let <span><math><mi>Σ</mi><mo>(</mo><mi>S</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∪</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></msubsup><msub><mrow><mi>Σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∪</mo></mrow><mrow><mi>t</mi><mo>=</mo><mi>k</mi></mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></msubsup><msub><mrow><mi>Σ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo></math></span>. It is known that if <span><math><mn>0</mn><mo>∉</mo><mi>Σ</mi><mo>(</mo><mi>S</mi><mo>)</mo></math></span>, then <span><math><mo>|</mo><mi>Σ</mi><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>+</mo><mo>|</mo><mrow><mi>supp</mi></mrow><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mo>−</mo><mn>1</mn></math></span>. In this paper, we determine the structure of a sequence <em>S</em> satisfying <span><math><mn>0</mn><mo>∉</mo><mi>Σ</mi><mo>(</mo><mi>S</mi><mo>)</mo></math></span> and <span><math><mo>|</mo><mi>Σ</mi><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mo>=</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>+</mo><mo>|</mo><mrow><mi>supp</mi></mrow><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mo>−</mo><mn>1</mn></math></span>. As a consequence, we can give a counterexample of a conjecture of Gao, Grynkiewicz, and Xia. Moreover, we prove that if <span><math><mo>|</mo><mi>S</mi><mo>|</mo><mo>&gt;</mo><mi>k</mi></math></span> and <span><math><mn>0</mn><mo>∉</mo><msub><mrow><mi>Σ</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>∪</mo><mrow><mi>supp</mi></mrow><mo>(</mo><mi>S</mi><mo>)</mo></math></span>, then <span><math><mo>|</mo><msub><mrow><mi>Σ</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>−</mo><mi>k</mi><mo>+</mo><mo>|</mo><mrow><mi>supp</mi></mrow><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo></math></span>. Then we can give an alternative proof of a conjecture of Hamidoune, which was first proved by Gao, Grynkiewicz, and Xia.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"218 ","pages":"Article 106124"},"PeriodicalIF":1.2,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145361864","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":"Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials","authors":"Tewodros Amdeberhan ,&nbsp;Leonid G. Fel ,&nbsp;Ken Ono","doi":"10.1016/j.jcta.2025.106123","DOIUrl":"10.1016/j.jcta.2025.106123","url":null,"abstract":"<div><div>Ramanujan derived a sequence of even weight 2<em>n</em> quasimodular forms <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> from derivatives of Jacobi's weight 3/2 theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of all nonnegative integer weights with minimal input: a weight 1 modular form and a power series <span><math><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. Using the weight 1 form <span><math><mi>θ</mi><msup><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>exp</mi><mo>⁡</mo><mo>(</mo><mi>X</mi><mo>/</mo><mn>2</mn><mo>)</mo></math></span>, we obtain a sequence <span><math><mo>{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>}</mo></math></span> of weight <em>n</em> quasimodular forms on <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>4</mn><mo>)</mo></math></span> whose symmetric function avatars <span><math><msub><mrow><mover><mrow><mi>Y</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> are the symmetric polynomials <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> that arise naturally in the study of syzygies of numerical semigroups. With this information, we settle two conjectures about the <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span>. Finally, we note that these polynomials are systematically given in terms of the Borel-Hirzebruch <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-genus for spin manifolds, where one identifies power sum symmetric functions <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> with Pontryagin classes.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"218 ","pages":"Article 106123"},"PeriodicalIF":1.2,"publicationDate":"2025-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145324044","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":"Dissection of the quintuple product, with applications","authors":"Tim Huber ,&nbsp;James Mc Laughlin ,&nbsp;Dongxi Ye","doi":"10.1016/j.jcta.2025.106122","DOIUrl":"10.1016/j.jcta.2025.106122","url":null,"abstract":"&lt;div&gt;&lt;div&gt;This work considers the &lt;em&gt;m&lt;/em&gt;-dissection (for &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;≢&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;) of the general quintuple product&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&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;msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&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;mi&gt;q&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&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;mi&gt;q&lt;/mi&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;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; Multiple novel applications arise from this &lt;em&gt;m&lt;/em&gt;-dissection. For example, we derive the general partition identity&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;24&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt; for all &lt;/mtext&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;≡&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a square-free positive integer relatively prime to 6; &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is defined, for &lt;em&gt;S&lt;/em&gt; the set of positive integers containing no multiples of &lt;em&gt;m&lt;/em&gt;, to be the number of partitions of &lt;em&gt;n&lt;/em&gt; into an &lt;u&gt;even&lt;/u&gt; number of distinct parts from &lt;em&gt;S&lt;/em&gt; minus the number of partitions of &lt;em&gt;n&lt;/em&gt; into an &lt;u&gt;odd&lt;/u&gt; number of distinct parts from &lt;em&gt;S&lt;/em&gt;; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; denotes the number of &lt;em&gt;m&lt;/em&gt;-regular partitions of &lt;em&gt;n&lt;/em&gt;. The dissections allow us to prove a conjecture of Hirschhorn concerning the &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;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;-dissection of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, as well as determine the pattern of the sign changes of the coefficients &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; of the infinite product&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&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;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mro","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"219 ","pages":"Article 106122"},"PeriodicalIF":1.2,"publicationDate":"2025-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321693","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":"Near triple arrays","authors":"Alexey Gordeev,&nbsp;Klas Markström,&nbsp;Lars-Daniel Öhman","doi":"10.1016/j.jcta.2025.106121","DOIUrl":"10.1016/j.jcta.2025.106121","url":null,"abstract":"<div><div>We introduce <em>near triple arrays</em> as binary row-column designs with at most two consecutive values for the replication numbers of symbols, for the intersection sizes of pairs of rows, pairs of columns and pairs of a row and a column. Near triple arrays form a common generalization of such well-studied classes of designs as triple arrays, (near) Youden rectangles and Latin squares.</div><div>We enumerate near triple arrays for a range of small parameter sets and show that they exist in the vast majority of the cases considered. As a byproduct, we obtain the first complete enumerations of <span><math><mn>6</mn><mo>×</mo><mn>10</mn></math></span> triple arrays on 15 symbols, <span><math><mn>7</mn><mo>×</mo><mn>8</mn></math></span> triple arrays on 14 symbols and <span><math><mn>5</mn><mo>×</mo><mn>16</mn></math></span> triple arrays on 20 symbols.</div><div>Next, we give several constructions for families of near triple arrays, and e.g. show that near triple arrays with 3 rows and at least 6 columns exist for any number of symbols. Finally, we investigate a duality between row and column intersection sizes of a row-column design, and covering numbers for pairs of symbols by rows and columns. These duality results are used to obtain necessary conditions for the existence of near triple arrays. This duality also provides a new unified approach to earlier results on triple arrays and balanced grids.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"219 ","pages":"Article 106121"},"PeriodicalIF":1.2,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223096","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":"k-Adjoint of hyperplane arrangements","authors":"Weikang Liang ,&nbsp;Suijie Wang ,&nbsp;Chengdong Zhao","doi":"10.1016/j.jcta.2025.106120","DOIUrl":"10.1016/j.jcta.2025.106120","url":null,"abstract":"<div><div>In this paper, we introduce the <em>k</em>-adjoint of a given hyperplane arrangement <span><math><mi>A</mi></math></span> associated with rank-<em>k</em> elements in the intersection lattice <span><math><mi>L</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, which generalizes the classical adjoint proposed by Bixby and Coullard. The <em>k</em>-adjoint of <span><math><mi>A</mi></math></span> induces a decomposition of the Grassmannian, which we call the <span><math><mi>A</mi></math></span>-adjoint decomposition. Inspired by the work of Gelfand, Goresky, MacPherson, and Serganova, we generalize the matroid decomposition and refined Schubert decomposition of the Grassmannian from the perspective of <span><math><mi>A</mi></math></span>. Furthermore, we prove that these three decompositions are exactly the same decomposition. A notable application involves providing a combinatorial classification of all the <em>k</em>-dimensional restrictions of <span><math><mi>A</mi></math></span>. Consequently, we establish the antitonicity of some combinatorial invariants, such as Whitney numbers of the first kind and the independence numbers.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"219 ","pages":"Article 106120"},"PeriodicalIF":1.2,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223094","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":"Improved asymptotics for moments of reciprocal sums for partitions into distinct parts","authors":"Kathrin Bringmann ,&nbsp;Byungchan Kim ,&nbsp;Eunmi Kim","doi":"10.1016/j.jcta.2025.106119","DOIUrl":"10.1016/j.jcta.2025.106119","url":null,"abstract":"<div><div>In this paper we strongly improve asymptotics for <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> (respectively <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>) which sums reciprocals (respectively squares of reciprocals) of parts throughout all the partitions of <em>n</em> into distinct parts. The methods required are much more involved than in the case of usual partitions since the generating functions are not modular and also do not possess product expansions.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"219 ","pages":"Article 106119"},"PeriodicalIF":1.2,"publicationDate":"2025-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145160110","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":"Dimension identities, almost self-conjugate partitions, and BGG complexes for Hermitian symmetric pairs","authors":"William Q. Erickson,&nbsp;Markus Hunziker","doi":"10.1016/j.jcta.2025.106118","DOIUrl":"10.1016/j.jcta.2025.106118","url":null,"abstract":"<div><div>An <em>almost self-conjugate</em> (ASC) partition has a Young diagram in which each arm along the diagonal is exactly one box longer than its corresponding leg. Classically, the ASC partitions and their conjugates appear in two of Littlewood's symmetric function identities. These identities can be viewed as Euler characteristics of BGG complexes of the trivial representation, for classical Hermitian symmetric pairs. In this paper, we consider partitions in which the arm–leg difference is an arbitrary constant <em>m</em>. By viewing these partitions as highest weights, we establish an infinite family of dimension identities 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. We then interpret this result in the context of blocks in parabolic category <span><math><mi>O</mi></math></span>: in particuar, we exhibit six infinite families of congruent blocks whose corresponding posets of highest weights consist of the partitions in question. These posets, in turn, lead to generalizations of the Littlewood identities and their corresponding BGG complexes. Our results in this paper shed light on the surprising combinatorics underlying the work of Enright and Willenbring (2004).</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"219 ","pages":"Article 106118"},"PeriodicalIF":1.2,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145128159","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 Schur polynomials in all primitive nth roots of unity","authors":"Masaki Hidaka,&nbsp;Minoru Itoh","doi":"10.1016/j.jcta.2025.106107","DOIUrl":"10.1016/j.jcta.2025.106107","url":null,"abstract":"<div><div>We show that the Schur polynomials in all primitive <em>n</em>th roots of unity are 1, 0, or −1, if <em>n</em> has at most two distinct odd prime factors. This result can be regarded as a generalization of properties of the coefficients of the cyclotomic polynomial and its multiplicative inverse. The key to the proof is the concept of a unimodular system of vectors. Namely, this result can be reduced to the unimodularity of the tensor product of two maximal circuits (here we call a vector system a maximal circuit, if it can be expressed as <span><math><mi>B</mi><mo>∪</mo><mo>{</mo><mo>−</mo><mo>∑</mo><mi>B</mi><mo>}</mo></math></span> with some basis <em>B</em>).</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"218 ","pages":"Article 106107"},"PeriodicalIF":1.2,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145094910","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":"Existences of semiregular automorphisms of edge-transitive graphs of odd prime valency","authors":"Wenjuan Luo,&nbsp;Jing Xu","doi":"10.1016/j.jcta.2025.106117","DOIUrl":"10.1016/j.jcta.2025.106117","url":null,"abstract":"<div><div>The Polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism whose cycles all have the same length. Similarly, in <span><span>[11]</span></span> the authors asked if every connected regular edge-transitive graph admits a semiregular automorphism. In this paper we prove that edge-transitive graphs of odd prime valency have a semiregular automorphism.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"218 ","pages":"Article 106117"},"PeriodicalIF":1.2,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145094909","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":"Arithmetic properties of MacMahon-type sums of divisors: The odd case","authors":"James A. Sellers ,&nbsp;Roberto Tauraso","doi":"10.1016/j.jcta.2025.106105","DOIUrl":"10.1016/j.jcta.2025.106105","url":null,"abstract":"<div><div>A century ago, P. A. MacMahon introduced two families of generating functions,<span><span><span><math><munder><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&lt;</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><mo>⋯</mo><mo>&lt;</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></munder><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup></mrow><mrow><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mspace></mspace><mtext> and </mtext><munder><mo>∑</mo><mrow><mtable><mtr><mtd><mn>1</mn><mo>≤</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&lt;</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><mo>⋯</mo><mo>&lt;</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mtext> odd</mtext></mrow></mtd></mtr></mtable></mrow></munder><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup></mrow><mrow><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>,</mo></math></span></span></span> which connect sum-of-divisors functions and integer partitions. These have recently drawn renewed attention. In particular, Amdeberhan, Andrews, and Tauraso extended the first family above by defining<span><span><span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>:</mo><mo>=</mo><munder><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&lt;</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><mo>⋯</mo><mo>&lt;</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></munder><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup></mrow><mrow><mn>1</mn><mo>+</mo><mi>a</mi><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><m","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"218 ","pages":"Article 106105"},"PeriodicalIF":1.2,"publicationDate":"2025-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924919","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}