Journal of Combinatorial Theory Series A最新文献

{"title":"On power monoids and their automorphisms","authors":"Salvatore Tringali,&nbsp;Weihao Yan","doi":"10.1016/j.jcta.2024.105961","DOIUrl":"10.1016/j.jcta.2024.105961","url":null,"abstract":"<div><div>Endowed with the binary operation of set addition, the family <span><math><msub><mrow><mi>P</mi></mrow><mrow><mrow><mi>fin</mi></mrow><mo>,</mo><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> of all finite subsets of <span><math><mi>N</mi></math></span> containing 0 forms a monoid, with the singleton {0} as its neutral element.</div><div>We show that the only non-trivial automorphism of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mrow><mi>fin</mi></mrow><mo>,</mo><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> is the involution <span><math><mi>X</mi><mo>↦</mo><mi>max</mi><mo>⁡</mo><mi>X</mi><mo>−</mo><mi>X</mi></math></span>. The proof leverages ideas from additive number theory and proceeds through an unconventional induction on what we call the boxing dimension of a finite set of integers, that is, the smallest number of (discrete) intervals whose union is the set itself.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105961"},"PeriodicalIF":0.9,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319544","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 non-empty cross-t-intersecting families","authors":"Anshui Li ,&nbsp;Huajun Zhang","doi":"10.1016/j.jcta.2024.105960","DOIUrl":"10.1016/j.jcta.2024.105960","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be families of <em>k</em>-element subsets of a <em>n</em>-element set. We call them cross-<em>t</em>-intersecting if <span><math><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>|</mo><mo>≥</mo><mi>t</mi></math></span> for any <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> with <span><math><mi>i</mi><mo>≠</mo><mi>j</mi></math></span>. In this paper we will prove that, for <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi><mo>−</mo><mi>t</mi><mo>+</mo><mn>1</mn></math></span>, if <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> are non-empty cross-<em>t</em>-intersecting families, then<span><span><span><math><munder><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>m</mi></mrow></munder><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>≤</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><munder><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>t</mi><mo>−</mo><mn>1</mn></mrow></munder><mrow><mo>(</mo><mtable><mtr><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mi>i</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mi>i</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>m</mi><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>}</mo><mo>,</mo></math></span></span></span> where <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> is the size of the maximum <em>t</em>-intersecting family of <span><math><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></math></span>. Moreover, the extremal families attaining the upper bound are characterized.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105960"},"PeriodicalIF":0.9,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142316250","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":"Avoiding intersections of given size in finite affine spaces AG(n,2)","authors":"Benedek Kovács ,&nbsp;Zoltán Lóránt Nagy","doi":"10.1016/j.jcta.2024.105959","DOIUrl":"10.1016/j.jcta.2024.105959","url":null,"abstract":"<div><div>We study the set of intersection sizes of a <em>k</em>-dimensional affine subspace and a point set of size <span><math><mi>m</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></math></span> of the <em>n</em>-dimensional binary affine space <span><math><mrow><mi>AG</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. Following the theme of Erdős, Füredi, Rothschild and T. Sós, we partially determine which local densities in <em>k</em>-dimensional affine subspaces are unavoidable in all <em>m</em>-element point sets in the <em>n</em>-dimensional affine space.</div><div>We also show constructions of point sets for which the intersection sizes with <em>k</em>-dimensional affine subspaces take values from a set of a small size compared to <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>. These are built up from affine subspaces and so-called subspace evasive sets. Meanwhile, we improve the best known upper bounds on subspace evasive sets and apply results concerning the canonical signed-digit (CSD) representation of numbers.</div><div><em>Keywords</em>: unavoidable, affine subspaces, evasive sets, random methods, canonical signed-digit number system.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105959"},"PeriodicalIF":0.9,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000980/pdfft?md5=62687b67d599290d3f204041642a9a6a&pid=1-s2.0-S0097316524000980-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314104","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 rank two Leonard pair in Terwilliger algebras of Doob graphs","authors":"John Vincent S. Morales","doi":"10.1016/j.jcta.2024.105958","DOIUrl":"10.1016/j.jcta.2024.105958","url":null,"abstract":"<div><div>Let <span><math><mi>Γ</mi><mo>=</mo><mi>Γ</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> denote the Doob graph formed by the Cartesian product of the <em>n</em>th Cartesian power of the Shrikhande graph and the <em>m</em>th Cartesian power of the complete graph on four vertices. Let <span><math><mi>T</mi><mo>=</mo><mi>T</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> denote the Terwilliger algebra of Γ with respect to a fixed vertex <em>x</em> of Γ and let <em>W</em> denote an arbitrary non-thin irreducible <em>T</em>-module in the standard module of Γ. In (Morales and Palma, 2021 <span><span>[25]</span></span>), it was shown that there exists a Lie algebra embedding <em>π</em> from the special orthogonal algebra <span><math><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> into <em>T</em> and that <em>W</em> is an irreducible <span><math><mi>π</mi><mo>(</mo><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo></math></span>-module. In this paper, we consider two Cartan subalgebras <span><math><mi>h</mi><mo>,</mo><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> of <span><math><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> such that <span><math><mi>h</mi><mo>,</mo><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> generate <span><math><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>. Using the embedding <span><math><mi>π</mi><mo>:</mo><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>→</mo><mi>T</mi></math></span>, we show that <span><math><mi>π</mi><mo>(</mo><mi>h</mi><mo>)</mo></math></span> and <span><math><mi>π</mi><mo>(</mo><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>)</mo></math></span> act on <em>W</em> as a rank two Leonard pair. We also obtain several direct sum decompositions of <em>W</em> akin to how split decompositions are obtained from Leonard pairs of rank one.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105958"},"PeriodicalIF":0.9,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312553","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":"Covering the set of p-elements in finite groups by proper subgroups","authors":"Attila Maróti ,&nbsp;Juan Martínez ,&nbsp;Alexander Moretó","doi":"10.1016/j.jcta.2024.105954","DOIUrl":"10.1016/j.jcta.2024.105954","url":null,"abstract":"<div><p>Let <em>p</em> be a prime and let <em>G</em> be a finite group which is generated by the set <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> of its <em>p</em>-elements. We show that if <em>G</em> is solvable and not a <em>p</em>-group, then the minimal number <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of proper subgroups of <em>G</em> whose union contains <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is equal to 1 less than the minimal number of proper subgroups of <em>G</em> whose union is <em>G</em>. For <em>p</em>-solvable groups <em>G</em>, we always have <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>p</mi><mo>+</mo><mn>1</mn></math></span>. We study the case of alternating and symmetric groups <em>G</em> in detail.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105954"},"PeriodicalIF":0.9,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000931/pdfft?md5=35b9a89a7b1644f6cad2cea930c20904&pid=1-s2.0-S0097316524000931-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271023","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":"Proofs of some conjectures of Merca on truncated series involving the Rogers-Ramanujan functions","authors":"Yongqiang Chen,&nbsp;Olivia X.M. Yao","doi":"10.1016/j.jcta.2024.105956","DOIUrl":"10.1016/j.jcta.2024.105956","url":null,"abstract":"<div><p>In 2012, Andrews and Merca investigated the truncated version of the Euler pentagonal number theorem. Their work has opened up a new study on truncated theta series and has inspired several mathematicians to work on the topic. In 2019, Merca studied the Rogers-Ramanujan functions and posed three groups of conjectures on truncated series involving the Rogers-Ramanujan functions. In this paper, we present a uniform method to prove the three groups of conjectures given by Merca based on a result due to Pólya and Szegö.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105956"},"PeriodicalIF":0.9,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142270376","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 the proportion of metric matroids whose Jacobians have nontrivial p-torsion","authors":"Sergio Ricardo Zapata Ceballos","doi":"10.1016/j.jcta.2024.105953","DOIUrl":"10.1016/j.jcta.2024.105953","url":null,"abstract":"<div><p>We study the proportion of metric matroids whose Jacobians have nontrivial <em>p</em>-torsion. We establish a correspondence between these Jacobians and the <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-rational points on configuration hypersurfaces, thereby relating their proportions. By counting points over finite fields, we prove that the proportion of these Jacobians is asymptotically equivalent to <span><math><mn>1</mn><mo>/</mo><mi>p</mi></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105953"},"PeriodicalIF":0.9,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S009731652400092X/pdfft?md5=d8e2893424d34795a1338a7aa80035a5&pid=1-s2.0-S009731652400092X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142243316","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":"Approximate generalized Steiner systems and near-optimal constant weight codes","authors":"Miao Liu ,&nbsp;Chong Shangguan","doi":"10.1016/j.jcta.2024.105955","DOIUrl":"10.1016/j.jcta.2024.105955","url":null,"abstract":"<div><p>Constant weight codes (CWCs) and constant composition codes (CCCs) are two important classes of codes that have been studied extensively in both combinatorics and coding theory for nearly sixty years. In this paper we show that for <em>all</em> fixed odd distances, there exist near-optimal CWCs and CCCs asymptotically achieving the classic Johnson-type upper bounds.</p><p>Let <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>w</mi><mo>)</mo></math></span> denote the maximum size of <em>q</em>-ary CWCs of length <em>n</em> with constant weight <em>w</em> and minimum distance <em>d</em>. One of our main results shows that for <em>all</em> fixed <span><math><mi>q</mi><mo>,</mo><mi>w</mi></math></span> and odd <em>d</em>, one has <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mo>⁡</mo><mfrac><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>t</mi></mrow></msup></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mi>w</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>)</mo></mrow></mfrac></math></span>, where <span><math><mi>t</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>w</mi><mo>−</mo><mi>d</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. This implies the existence of near-optimal generalized Steiner systems originally introduced by Etzion, and can be viewed as a counterpart of a celebrated result of Rödl on the existence of near-optimal Steiner systems. Note that prior to our work, very little is known about <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>w</mi><mo>)</mo></math></span> for <span><math><mi>q</mi><mo>≥</mo><mn>3</mn></math></span>. A similar result is proved for the maximum size of CCCs.</p><p>We provide different proofs for our two main results, based on two strengthenings of the well-known Frankl-Rödl-Pippenger theorem on the existence of near-optimal matchings in hypergraphs: the first proof follows by Kahn's linear programming variation of the above theorem, and the second follows by the recent independent work of Delcourt-Postle, and Glock-Joos-Kim-Kühn-Lichev on the existence of near-optimal matchings avoiding certain forbidden configurations.</p><p>We also present several intriguing open questions for future research.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105955"},"PeriodicalIF":0.9,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000943/pdfft?md5=65eb96db9a426be78f5105ffe48c2ece&pid=1-s2.0-S0097316524000943-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142168936","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 note on tournament m-semiregular representations of finite groups","authors":"Jia-Li Du","doi":"10.1016/j.jcta.2024.105952","DOIUrl":"10.1016/j.jcta.2024.105952","url":null,"abstract":"<div><p>For a positive integer <em>m</em>, a group <em>G</em> is said to admit a <em>tournament m-semiregular representation</em> (T<em>m</em>SR for short) if there exists a tournament Γ such that the automorphism group of Γ is isomorphic to <em>G</em> and acts semiregularly on the vertex set of Γ with <em>m</em> orbits. It is easy to see that every finite group of even order does not admit a T<em>m</em>SR for any positive integer <em>m</em>. The T1SR is the well-known tournament regular representation (TRR for short). In 1970s, Babai and Imrich proved that every finite group of odd order admits a TRR except for <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span>, and every group (finite or infinite) without element of order 2 having an independent generating set admits a T2SR in (1979) <span><span>[3]</span></span>. Later, Godsil correct the result by showing that the only finite groups of odd order without a TRR are <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> by a probabilistic approach in (1986) <span><span>[11]</span></span>. In this note, it is shown that every finite group of odd order has a T<em>m</em>SR for every <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105952"},"PeriodicalIF":0.9,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000918/pdfft?md5=9f9703a561ce567e377942546fcc91e2&pid=1-s2.0-S0097316524000918-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142137277","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 separating Noether number of abelian groups of rank two","authors":"Barna Schefler","doi":"10.1016/j.jcta.2024.105951","DOIUrl":"10.1016/j.jcta.2024.105951","url":null,"abstract":"<div><p>The exact value of the separating Noether number of an arbitrary finite abelian group of rank two is determined. This is done by a detailed study of the monoid of zero-sum sequences over the group.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"209 ","pages":"Article 105951"},"PeriodicalIF":0.9,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000906/pdfft?md5=440ee51944dfcc0609d997a9aaba1b3f&pid=1-s2.0-S0097316524000906-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142095181","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}