Journal of Combinatorial Theory Series A最新文献

{"title":"The second largest eigenvalue of normal Cayley graphs on symmetric groups generated by cycles","authors":"Yuxuan Li,&nbsp;Binzhou Xia,&nbsp;Sanming Zhou","doi":"10.1016/j.jcta.2024.105885","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105885","url":null,"abstract":"<div><p>We study the normal Cayley graphs <span><math><mrow><mi>Cay</mi></mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>I</mi><mo>)</mo><mo>)</mo></math></span> on the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, where <span><math><mi>I</mi><mo>⊆</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> and <span><math><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>I</mi><mo>)</mo></math></span> is the set of all cycles in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with length in <em>I</em>. We prove that the strictly second largest eigenvalue of <span><math><mrow><mi>Cay</mi></mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>I</mi><mo>)</mo><mo>)</mo></math></span> can only be achieved by at most four irreducible representations of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and we determine further the multiplicity of this eigenvalue in several special cases. As a corollary, in the case when <em>I</em> contains neither <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> nor <em>n</em> we know exactly when <span><math><mrow><mi>Cay</mi></mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>I</mi><mo>)</mo><mo>)</mo></math></span> has the Aldous property, namely the strictly second largest eigenvalue is attained by the standard representation of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and we obtain that <span><math><mrow><mi>Cay</mi></mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>I</mi><mo>)</mo><mo>)</mo></math></span> does not have the Aldous property whenever <span><math><mi>n</mi><mo>∈</mo><mi>I</mi></math></span>. As another corollary of our main results, we prove a recent conjecture on the second largest eigenvalue of <span><math><mrow><mi>Cay</mi></mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><mi>k</mi><mo>}</mo><mo>)</mo><mo>)</mo></math></span> where <span><math><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>2</mn></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105885"},"PeriodicalIF":1.1,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000244/pdfft?md5=f945a709cc7931a6640e76d02ea647ea&pid=1-s2.0-S0097316524000244-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140014637","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 short combinatorial proof of dimension identities of Erickson and Hunziker","authors":"Nishu Kumari","doi":"10.1016/j.jcta.2024.105883","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105883","url":null,"abstract":"<div><p>In a recent paper (<span>arXiv:2301.09744</span><svg><path></path></svg>), Erickson and Hunziker consider partitions in which the arm–leg difference is an arbitrary constant <em>m</em>. In previous works, these partitions are called <span><math><mo>(</mo><mo>−</mo><mi>m</mi><mo>)</mo></math></span>-asymmetric partitions. Regarding these partitions and their conjugates as highest weights, they prove an identity yielding an infinite family of dimension equalities 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. Their proof proceeds by the manipulations of the hook content formula. We give a simple combinatorial proof of their result.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105883"},"PeriodicalIF":1.1,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139998969","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 deepest cycle of a random mapping","authors":"Ljuben Mutafchiev ,&nbsp;Steven Finch","doi":"10.1016/j.jcta.2024.105875","DOIUrl":"10.1016/j.jcta.2024.105875","url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the set of all mappings <span><math><mi>T</mi><mo>:</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo><mo>→</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. The corresponding graph of <em>T</em> is a union of disjoint connected unicyclic components. We assume that each <span><math><mi>T</mi><mo>∈</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is chosen uniformly at random (i.e., with probability <span><math><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>n</mi></mrow></msup></math></span>). The cycle of <em>T</em> contained within its largest component is called the <em>deepest</em> one. For any <span><math><mi>T</mi><mo>∈</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, let <span><math><msub><mrow><mi>ν</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math></span> denote the length of this cycle. In this paper, we establish the convergence in distribution of <span><math><msub><mrow><mi>ν</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></math></span> and find the limits of its expectation and variance as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>. For <em>n</em> large enough, we also show that nearly 55% of all cyclic vertices of a random mapping <span><math><mi>T</mi><mo>∈</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> lie in its deepest cycle and that a vertex from the longest cycle of <em>T</em> does not belong to its largest component with approximate probability 0.075.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105875"},"PeriodicalIF":1.1,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139937806","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":"Two conjectures of Andrews, Merca and Yee on truncated theta series","authors":"Shane Chern ,&nbsp;Ernest X.W. Xia","doi":"10.1016/j.jcta.2024.105874","DOIUrl":"10.1016/j.jcta.2024.105874","url":null,"abstract":"<div><p>In their study of the truncation of Euler's pentagonal number theorem, Andrews and Merca introduced a partition function <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> to count the number of partitions of <em>n</em> in which <em>k</em> is the least integer that is not a part and there are more parts exceeding <em>k</em> than there are below <em>k</em>. In recent years, two conjectures concerning <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> on truncated theta series were posed by Andrews, Merca, and Yee. In this paper, we prove that the two conjectures are true for sufficiently large <em>n</em> whenever <em>k</em> is fixed.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105874"},"PeriodicalIF":1.1,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139937815","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":"Constructing generalized Heffter arrays via near alternating sign matrices","authors":"L. Mella ,&nbsp;T. Traetta","doi":"10.1016/j.jcta.2024.105873","DOIUrl":"10.1016/j.jcta.2024.105873","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;em&gt;S&lt;/em&gt; be a subset of a group &lt;em&gt;G&lt;/em&gt; (not necessarily abelian) such that &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is empty or contains only elements of order 2, and let &lt;span&gt;&lt;math&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;h&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;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;h&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;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&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;mn&gt;1&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;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&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;. A &lt;em&gt;generalized Heffter array&lt;/em&gt; GHA&lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; over &lt;em&gt;G&lt;/em&gt; is an &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; matrix &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&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;i&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; such that: the &lt;em&gt;i&lt;/em&gt;-th row (resp. &lt;em&gt;j&lt;/em&gt;-th column) of &lt;em&gt;A&lt;/em&gt; contains exactly &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;h&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; (resp. &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;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;) nonzero elements, and the list &lt;span&gt;&lt;math&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;i&lt;/mi&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;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;mi&gt;j&lt;/mi&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;mi&gt;i&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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; equals &lt;em&gt;λ&lt;/em&gt; times the set &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of &lt;em&gt;A&lt;/em&gt; sums to zero (resp. a nonzero element), with respect to some ordering.&lt;/p&gt;&lt;p&gt;In this paper, we use &lt;em&gt;near alternating sign matrices&lt;/em&gt; to build both zero and nonzero sum GHAs, over cyclic groups, having the further strong property of being simple. In particular, we construct zero sum and simple GHAs whose row and column weights are congruent to 0 modulo 4. This result also provides the first infinite family of simple (classic) Heffter arrays to be rectangular (&lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;) and with less than &lt;em&gt;n&lt;/em&gt; nonzero entries in each row. Furthermore, we build nonzero sum GHA&lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; over an arbitrary group &lt;em&gt;G&lt;/em&gt; whenever &lt;em&gt;S&lt;/em&gt; contains enough noninvolutions, th","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105873"},"PeriodicalIF":1.1,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139916880","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 maximal number of elements pairwise generating the finite alternating group","authors":"Francesco Fumagalli ,&nbsp;Martino Garonzi ,&nbsp;Pietro Gheri","doi":"10.1016/j.jcta.2024.105870","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105870","url":null,"abstract":"<div><p>Let <em>G</em> be the alternating group of degree <em>n</em>. Let <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the maximal size of a subset <em>S</em> of <em>G</em> such that <span><math><mo>〈</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>〉</mo><mo>=</mo><mi>G</mi></math></span> whenever <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi></math></span> and <span><math><mi>x</mi><mo>≠</mo><mi>y</mi></math></span> and let <span><math><mi>σ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the minimal size of a family of proper subgroups of <em>G</em> whose union is <em>G</em>. We prove that, when <em>n</em> varies in the family of composite numbers, <span><math><mi>σ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>/</mo><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> tends to 1 as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>. Moreover, we explicitly calculate <span><math><mi>σ</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>21</mn></math></span> congruent to 3 modulo 18.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105870"},"PeriodicalIF":1.1,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000098/pdfft?md5=0f713e308f01065a0eed53c25b2ba78c&pid=1-s2.0-S0097316524000098-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139732729","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":"Most plane curves over finite fields are not blocking","authors":"Shamil Asgarli ,&nbsp;Dragos Ghioca ,&nbsp;Chi Hoi Yip","doi":"10.1016/j.jcta.2024.105871","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105871","url":null,"abstract":"<div><p>A plane curve <span><math><mi>C</mi><mo>⊂</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> of degree <em>d</em> is called <em>blocking</em> if every <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-line in the plane meets <em>C</em> at some <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-point. We prove that the proportion of blocking curves among those of degree <em>d</em> is <span><math><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> when <span><math><mi>d</mi><mo>≥</mo><mn>2</mn><mi>q</mi><mo>−</mo><mn>1</mn></math></span> and <span><math><mi>q</mi><mo>→</mo><mo>∞</mo></math></span>. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition <span><math><mi>d</mi><mo>≥</mo><mn>3</mn><mi>p</mi></math></span> and <span><math><mi>d</mi><mo>,</mo><mi>q</mi><mo>→</mo><mo>∞</mo></math></span>. Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-roots of random polynomials, we find that the limiting distribution of the number of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-points in the intersection of a random plane curve and a fixed <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-points contained in a union of <em>k</em> lines for <span><math><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"204 ","pages":"Article 105871"},"PeriodicalIF":1.1,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139719451","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 Q-polynomial structure for the Attenuated Space poset Aq(N,M)","authors":"Paul Terwilliger","doi":"10.1016/j.jcta.2024.105872","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105872","url":null,"abstract":"&lt;div&gt;&lt;p&gt;The goal of this article is to display a &lt;em&gt;Q&lt;/em&gt;-polynomial structure for the Attenuated Space poset &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;q&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;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. The poset &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;q&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;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is briefly described as follows. Start with an &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-dimensional vector space &lt;em&gt;H&lt;/em&gt; over a finite field with &lt;em&gt;q&lt;/em&gt; elements. Fix an &lt;em&gt;M&lt;/em&gt;-dimensional subspace &lt;em&gt;h&lt;/em&gt; of &lt;em&gt;H&lt;/em&gt;. The vertex set &lt;em&gt;X&lt;/em&gt; of &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;q&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;mi&gt;M&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; consists of the subspaces of &lt;em&gt;H&lt;/em&gt; that have zero intersection with &lt;em&gt;h&lt;/em&gt;. The partial order on &lt;em&gt;X&lt;/em&gt; is the inclusion relation. The &lt;em&gt;Q&lt;/em&gt;-polynomial structure involves two matrices &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Mat&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with the following entries. For &lt;span&gt;&lt;math&gt;&lt;mi&gt;y&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;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; the matrix &lt;em&gt;A&lt;/em&gt; has &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-entry 1 (if &lt;em&gt;y&lt;/em&gt; covers &lt;em&gt;z&lt;/em&gt;); &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;dim&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; (if &lt;em&gt;z&lt;/em&gt; covers &lt;em&gt;y&lt;/em&gt;); and 0 (if neither of &lt;span&gt;&lt;math&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; covers the other). The matrix &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; is diagonal, with &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-entry &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;dim&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; for all &lt;span&gt;&lt;math&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. By construction, &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; has &lt;span&gt;&lt;math&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; eigenspaces. By construction, &lt;em&gt;A&lt;/em&gt; acts on these eigenspaces in a (block) tridiagonal fashion. We show that &lt;em&gt;A&lt;/em&gt; is diagonalizable, with &lt;span&gt;&lt;math&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; eigenspaces. We show that &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; acts on these eigenspaces in a (block) tridiagonal fashion. Using this action, we show that &lt;em&gt;A&lt;/em&gt; is &lt;em&gt;Q&lt;/em&gt;-polynomial. We show that &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; satisfy a pair of relations called the tridiagonal relations. We consider the subalgebra &lt;em&gt;T&lt;/em&gt; of ","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105872"},"PeriodicalIF":1.1,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139713906","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":"Spectral characterization of the complete graph removing a cycle","authors":"Muhuo Liu ,&nbsp;Xiaofeng Gu ,&nbsp;Haiying Shan ,&nbsp;Zoran Stanić","doi":"10.1016/j.jcta.2024.105868","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105868","url":null,"abstract":"<div><p>A graph is determined by its spectrum if there is not another graph with the same spectrum. Cámara and Haemers proved that the graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, obtained from the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <em>n</em> vertices by deleting all edges of a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> with <em>k</em> vertices, is determined by its spectrum for <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></math></span>, but not for <span><math><mi>k</mi><mo>=</mo><mn>6</mn></math></span>. In this paper, we show that <span><math><mi>k</mi><mo>=</mo><mn>6</mn></math></span> is the unique exception for the spectral determination of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105868"},"PeriodicalIF":1.1,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139713905","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 divisor class group of a discrete polymatroid","authors":"Jürgen Herzog ,&nbsp;Takayuki Hibi ,&nbsp;Somayeh Moradi ,&nbsp;Ayesha Asloob Qureshi","doi":"10.1016/j.jcta.2024.105869","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105869","url":null,"abstract":"<div><p>In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete polymatroid, its toric ring is studied deeply for several classes of polymatroids.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"205 ","pages":"Article 105869"},"PeriodicalIF":1.1,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139709553","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}