Journal of Combinatorial Theory Series A最新文献

{"title":"On joint short minimal zero-sum subsequences over finite abelian groups of rank two","authors":"Yushuang Fan ,&nbsp;Qinghai Zhong","doi":"10.1016/j.jcta.2024.105984","DOIUrl":"10.1016/j.jcta.2024.105984","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>)</mo></math></span> be a finite abelian group and let <span><math><msup><mrow><mi>η</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the smallest integer <em>ℓ</em> such that every sequence over <span><math><mi>G</mi><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span> of length <em>ℓ</em> has two joint short minimal zero-sum subsequences. In 2013, Gao et al. obtained that <span><math><msup><mrow><mi>η</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⊕</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>3</mn><mi>n</mi><mo>+</mo><mn>1</mn></math></span> for every <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> and solved the corresponding inverse problem for groups <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>⊕</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, where <em>p</em> is a prime. In this paper, we determine the precise value of <span><math><msup><mrow><mi>η</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for all finite abelian groups of rank 2 and resolve the corresponding inverse problem for groups <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⊕</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, where <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, which confirms a conjecture of Gao, Geroldinger and Wang for all <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> except <span><math><mi>n</mi><mo>=</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 105984"},"PeriodicalIF":0.9,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790045","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":"The degree of functions in the Johnson and q-Johnson schemes","authors":"Michael Kiermaier ,&nbsp;Jonathan Mannaert ,&nbsp;Alfred Wassermann","doi":"10.1016/j.jcta.2024.105979","DOIUrl":"10.1016/j.jcta.2024.105979","url":null,"abstract":"<div><div>In 1982, Cameron and Liebler investigated certain <em>special sets of lines</em> in <span><math><mi>PG</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, and gave several equivalent characterizations. Due to their interesting geometric and algebraic properties, these <em>Cameron-Liebler line classes</em> got much attention. Several generalizations and variants have been considered in the literature, the main directions being a variation of the dimensions of the involved spaces, and studying the analogous situation in the subset lattice. An important tool is the interpretation of the objects as Boolean functions in the <em>Johnson</em> and <em>q-Johnson schemes</em>.</div><div>In this article, we develop a unified theory covering all these variations. Generalized versions of algebraic and geometric properties will be investigated, having a parallel in the notion of <em>designs</em> and <em>antidesigns</em> in association schemes. This leads to a natural definition of the <em>degree</em> and the <em>weights</em> of functions in the ambient scheme, refining the existing definitions. We will study the effect of dualization and of elementary modifications of the ambient space on the degree and the weights. Moreover, a divisibility property of the sizes of Boolean functions of degree <em>t</em> will be proven.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 105979"},"PeriodicalIF":0.9,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746446","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":"Sequence reconstruction problem for deletion channels: A complete asymptotic solution","authors":"Van Long Phuoc Pham ,&nbsp;Keshav Goyal ,&nbsp;Han Mao Kiah","doi":"10.1016/j.jcta.2024.105980","DOIUrl":"10.1016/j.jcta.2024.105980","url":null,"abstract":"<div><div>Transmit a codeword <figure><img></figure>, that belongs to an <span><math><mo>(</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-deletion-correcting code of length <em>n</em>, over a <em>t</em>-deletion channel for some <span><math><mn>1</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>t</mi><mo>&lt;</mo><mi>n</mi></math></span>. Levenshtein (2001) <span><span>[10]</span></span>, proposed the problem of determining <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span>, the minimum number of distinct channel outputs required to uniquely reconstruct <figure><img></figure>. Prior to this work, <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> is known only when <span><math><mi>ℓ</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></math></span>. Here, we provide an asymptotically exact solution for all values of <em>ℓ</em> and <em>t</em>. Specifically, we show that <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>ℓ</mi></mrow></mtd></mtr><mtr><mtd><mi>ℓ</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mi>ℓ</mi><mo>)</mo><mo>!</mo></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mi>t</mi><mo>−</mo><mi>ℓ</mi></mrow></msup><mo>−</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>t</mi><mo>−</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span>. In the special instances: where <span><math><mi>ℓ</mi><mo>=</mo><mi>t</mi></math></span>, we show that <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>ℓ</mi></mrow></mtd></mtr><mtr><mtd><mi>ℓ</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span>; and when <span><math><mi>ℓ</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><mi>t</mi><mo>=</mo><mn>4</mn></math></span>, we show that <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>≤</mo><mn>20</mn><mi>n</mi><mo>−</mo><mn>150</mn></math></span>. We also provide a conjecture on the exact value of <span><math><mi>N</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> for all values of <em>n</em>, <em>ℓ</em>, and <em>t</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105980"},"PeriodicalIF":0.9,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720012","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":"Non-empty pairwise cross-intersecting families","authors":"Yang Huang,&nbsp;Yuejian Peng","doi":"10.1016/j.jcta.2024.105981","DOIUrl":"10.1016/j.jcta.2024.105981","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Two families &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; are cross-intersecting if &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mo&gt;∅&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for any &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. We call &lt;em&gt;t&lt;/em&gt; families &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;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;A&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;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; pairwise cross-intersecting families if &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;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &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;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are cross-intersecting for &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Additionally, if &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;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mo&gt;∅&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for each &lt;span&gt;&lt;math&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, then we say that &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;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;A&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;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are non-empty pairwise cross-intersecting. Let &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;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&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;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&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;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be non-empty pairwise cross-intersecting families with &lt;span&gt;&lt;math&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&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;k&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;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&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;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &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;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;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/m","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105981"},"PeriodicalIF":0.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720004","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 classification of the flag-transitive 2-(v,k,2) designs","authors":"Hongxue Liang ,&nbsp;Alessandro Montinaro","doi":"10.1016/j.jcta.2024.105983","DOIUrl":"10.1016/j.jcta.2024.105983","url":null,"abstract":"<div><div>In this paper, we provide a complete classification of 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span> designs admitting a flag-transitive automorphism group of affine type with the only exception of the semilinear 1-dimensional group. Alongside this analysis, we provide a construction of seven new families of such flag-transitive 2-designs, one of them infinite, and some of them involving remarkable objects such as <em>t</em>-spreads, translation planes, quadrics and Segre varieties.</div><div>Our result together with those of Alavi et al. <span><span>[1]</span></span>, <span><span>[2]</span></span>, Praeger et al. <span><span>[17]</span></span>, Zhou and the first author <span><span>[39]</span></span>, <span><span>[40]</span></span> provides a complete classification of 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span> design admitting a flag-transitive automorphism group with the only exception of the semilinear 1-dimensional case.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105983"},"PeriodicalIF":0.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701472","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":"Distributions of reciprocal sums of parts in integer partitions","authors":"Byungchan Kim ,&nbsp;Eunmi Kim","doi":"10.1016/j.jcta.2024.105982","DOIUrl":"10.1016/j.jcta.2024.105982","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the set of partitions of <em>n</em> into distinct parts, and <span><math><mi>srp</mi><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> be the sum of reciprocals of the parts of the partition <em>λ</em>. We show that as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>,<span><span><span><math><mi>E</mi><mo>[</mo><mi>srp</mi><mo>(</mo><mi>λ</mi><mo>)</mo><mo>:</mo><mi>λ</mi><mo>∈</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo><mo>∼</mo><mfrac><mrow><mi>log</mi><mo>⁡</mo><mo>(</mo><mn>3</mn><mi>n</mi><mo>)</mo></mrow><mrow><mn>4</mn></mrow></mfrac><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>Var</mi><mo>[</mo><mi>srp</mi><mo>(</mo><mi>λ</mi><mo>)</mo><mo>:</mo><mi>λ</mi><mo>∈</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo><mo>∼</mo><mfrac><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>24</mn></mrow></mfrac><mo>.</mo></math></span></span></span> Moreover, for <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the set of ordinary partitions of <em>n</em>, we show that as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>,<span><span><span><math><mi>E</mi><mo>[</mo><mi>srp</mi><mo>(</mo><mi>λ</mi><mo>)</mo><mo>:</mo><mi>λ</mi><mo>∈</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo><mo>∼</mo><mi>π</mi><msqrt><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></msqrt><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>Var</mi><mo>[</mo><mi>srp</mi><mo>(</mo><mi>λ</mi><mo>)</mo><mo>:</mo><mi>λ</mi><mo>∈</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo><mo>∼</mo><mfrac><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>15</mn></mrow></mfrac><mi>n</mi><mo>.</mo></math></span></span></span> To prove these asymptotic formulas in a uniform manner, we derive a general asymptotic formula using Wright's circle method.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105982"},"PeriodicalIF":0.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720007","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":"Dominance complexes, neighborhood complexes and combinatorial Alexander duals","authors":"Takahiro Matsushita ,&nbsp;Shun Wakatsuki","doi":"10.1016/j.jcta.2024.105978","DOIUrl":"10.1016/j.jcta.2024.105978","url":null,"abstract":"<div><div>We show that the dominance complex <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> coincides with the combinatorial Alexander dual of the neighborhood complex <span><math><mi>N</mi><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover><mo>)</mo></math></span> of the complement of <em>G</em>. Using this, we obtain a relation between the chromatic number <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of <em>G</em> and the homology group of <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. We also obtain several known results related to dominance complexes from well-known facts of neighborhood complexes. After that, we suggest a new method for computing the homology groups of the dominance complexes, using independence complexes of simple graphs. We show that several known computations of homology groups of dominance complexes can be reduced to known computations of independence complexes. Finally, we determine the homology group of <span><math><mi>D</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span> by determining the homotopy types of the independence complex of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105978"},"PeriodicalIF":0.9,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652948","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":"Upper bounds for the number of substructures in finite geometries from the container method","authors":"Sam Mattheus,&nbsp;Geertrui Van de Voorde","doi":"10.1016/j.jcta.2024.105968","DOIUrl":"10.1016/j.jcta.2024.105968","url":null,"abstract":"<div><div>We use techniques from algebraic and extremal combinatorics to derive upper bounds on the number of independent sets in several (hyper)graphs arising from finite geometry. In this way, we obtain asymptotically sharp upper bounds for partial ovoids and EKR-sets of flags in polar spaces, line spreads in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mi>r</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> and plane spreads in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, and caps in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>. The latter result extends work due to Roche-Newton and Warren <span><span>[21]</span></span> and Bhowmick and Roche-Newton <span><span>[6]</span></span>.</div><div>Finally, we investigate caps in <em>p</em>-random subsets of <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, which parallels recent work for arcs in projective planes by Bhowmick and Roche-Newton, and Roche-Newton and Warren <span><span>[6]</span></span>, <span><span>[21]</span></span>, and arcs in projective spaces by Chen, Liu, Nie and Zeng <span><span>[8]</span></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105968"},"PeriodicalIF":0.9,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593647","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 vector space generated by permutations of a trade or a design","authors":"E. Ghorbani ,&nbsp;S. Kamali ,&nbsp;G.B. Khosrovshahi","doi":"10.1016/j.jcta.2024.105969","DOIUrl":"10.1016/j.jcta.2024.105969","url":null,"abstract":"<div><div>Motivated by a classical result of Graver and Jurkat (1973) and Graham, Li, and Li (1980) in combinatorial design theory, which states that the permutations of <em>t</em>-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> minimal trades generate the vector space of all <em>t</em>-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> trades, we investigate the vector space spanned by permutations of an arbitrary trade. We prove that this vector space possesses a decomposition as a direct sum of subspaces formed in the same way by a specific family of so-called total trades. As an application, we demonstrate that for any <em>t</em>-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> design, its permutations can span the vector space generated by all <em>t</em>-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> designs for sufficiently large values of <em>v</em>. In other words, any <em>t</em>-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> design, or even any <em>t</em>-trade, can be expressed as a linear combination of permutations of a fixed <em>t</em>-design. This substantially extends a result by Ghodrati (2019), who proved the same result for Steiner designs.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105969"},"PeriodicalIF":0.9,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593648","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":"Some conjectures of Ballantine and Merca on truncated sums and the minimal excludant in congruences classes","authors":"Olivia X.M. Yao","doi":"10.1016/j.jcta.2024.105967","DOIUrl":"10.1016/j.jcta.2024.105967","url":null,"abstract":"<div><div>In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Since then, a number of results on truncated theta series have been proved. In this paper, we find the connections between truncated sums of certain partition functions and the minimal excludant statistic which has been found to exhibit connections with a handful of objects such as Dyson's crank. We present a uniform method to confirm five conjectures on truncated sums of certain partition functions given by Ballantine and Merca. In particular, we provide partition-theoretic interpretations for some truncated sums by using the minimal excludant in congruences classes.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105967"},"PeriodicalIF":0.9,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586416","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}