Journal of Combinatorial Theory Series A最新文献

{"title":"q-Analogs of divisible design graphs and Deza graphs","authors":"Dean Crnković ,&nbsp;Maarten De Boeck ,&nbsp;Francesco Pavese ,&nbsp;Andrea Švob","doi":"10.1016/j.jcta.2025.106047","DOIUrl":"10.1016/j.jcta.2025.106047","url":null,"abstract":"<div><div>Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and Meulenberg. In this paper, we introduce the notion of <em>q</em>-analogs of divisible design graphs and show that all <em>q</em>-analogs of divisible design graphs come from spreads, and are actually <em>q</em>-analogs of strongly regular graphs.</div><div>Deza graphs were introduced by Erickson, Fernando, Haemers, Hardy and Hemmeter in 1999. In this paper, we introduce <em>q</em>-analogs of Deza graphs. Further, we determine possible parameters, give examples of <em>q</em>-analogs of Deza graphs and characterize all non-strongly regular <em>q</em>-analogs of Deza graphs with the smallest parameters.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106047"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761107","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":"Exceptional 2-to-1 rational functions","authors":"Zhiguo Ding ,&nbsp;Michael E. Zieve","doi":"10.1016/j.jcta.2025.106046","DOIUrl":"10.1016/j.jcta.2025.106046","url":null,"abstract":"<div><div>For each odd prime power <em>q</em>, we describe a class of rational functions <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with the following unusual property: for every odd <em>j</em>, the function induced by <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>j</mi></mrow></msup></mrow></msub><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span> is 2-to-1. We also show that, among all known rational functions <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> which are 2-to-1 on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>j</mi></mrow></msup></mrow></msub><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span> for infinitely many <em>j</em>, our new functions are the only ones which cannot be written as compositions of rational functions of degree at most four, monomials, Dickson polynomials, and additive (linearized) polynomials.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106046"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761106","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":"Parity statistics on restricted permutations and the Catalan–Schett polynomials","authors":"Zhicong Lin ,&nbsp;Jing Liu ,&nbsp;Sherry H.F. Yan","doi":"10.1016/j.jcta.2025.106049","DOIUrl":"10.1016/j.jcta.2025.106049","url":null,"abstract":"<div><div>Motivated by Kitaev and Zhang's recent work on non-overlapping ascents in stack-sortable permutations and Dumont's permutation interpretation of the Jacobi elliptic functions, we investigate some parity statistics on restricted permutations. Some new related bijections are constructed and two refinements of the generating function for descents over 321-avoiding permutations due to Barnabei, Bonetti and Silimbanian are obtained. In particular, an open problem of Kitaev and Zhang about non-overlapping ascents on 321-avoiding permutations is solved and several combinatorial interpretations for the Catalan–Schett polynomials are found. The stack-sortable permutations are at the heart of our approaches.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106049"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761108","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 central limit theorem for a card shuffling problem","authors":"Shane Chern ,&nbsp;Lin Jiu ,&nbsp;Italo Simonelli","doi":"10.1016/j.jcta.2025.106048","DOIUrl":"10.1016/j.jcta.2025.106048","url":null,"abstract":"<div><div>Given a positive integer <em>n</em>, consider a permutation of <em>n</em> objects chosen uniformly at random. In this permutation, we collect maximal subsequences consisting of consecutive numbers arranged in ascending order called blocks. Each block is then merged, and after all merges, the elements of this new set are relabeled from 1 to the current number of elements. We continue to permute and merge this new set uniformly at random until only one object is left. In this paper, we investigate the distribution of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the number of permutations needed for this process to end. In particular, we find explicit asymptotic expressions for the mean value <span><math><mi>E</mi><mo>[</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span>, the variance <span><math><mrow><mi>Var</mi></mrow><mo>[</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span>, and higher central moments, and show that <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> satisfies a central limit theorem.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106048"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759116","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":"Normal edge-transitive Cayley graphs on non-abelian simple groups","authors":"Xing Zhang,&nbsp;Yan-Quan Feng,&nbsp;Fu-Gang Yin,&nbsp;Jin-Xin Zhou","doi":"10.1016/j.jcta.2025.106050","DOIUrl":"10.1016/j.jcta.2025.106050","url":null,"abstract":"<div><div>Let <em>Γ</em> be a Cayley graph on a finite group <em>G</em>, and let <span><math><msub><mrow><mi>N</mi></mrow><mrow><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> be the normalizer of <span><math><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> (the right regular representation of <em>G</em>) in the full automorphism group <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> of <em>Γ</em>. We say that <em>Γ</em> is a normal Cayley graph on <em>G</em> if <span><math><msub><mrow><mi>N</mi></mrow><mrow><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>=</mo><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span>, and that <em>Γ</em> is a normal edge-transitive Cayley graph on <em>G</em> if <span><math><msub><mrow><mi>N</mi></mrow><mrow><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> acts transitively on the edge set of <em>Γ</em>. In 1999, Praeger proved that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency 3 is normal. As an extension of this, in this paper, we prove that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency <em>p</em> is normal for each prime <em>p</em>. This, however, is not true for composite valency. We give a method to construct connected normal edge-transitive but non-normal Cayley graphs of certain groups, and using this, we prove that if <em>G</em> is either <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> for an odd prime <span><math><mi>q</mi><mo>≥</mo><mn>5</mn></math></span>, or <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>, then there exists a connected normal edge-transitive but non-normal 8-valent Cayley graph of <em>G</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106050"},"PeriodicalIF":0.9,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739311","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":"Finite versions of the Andrews–Gordon identity and Bressoud's identity","authors":"Heng Huat Chan ,&nbsp;Song Heng Chan","doi":"10.1016/j.jcta.2025.106035","DOIUrl":"10.1016/j.jcta.2025.106035","url":null,"abstract":"<div><div>In this article, we discuss finite versions of Euler's pentagonal number identity, the Rogers-Ramanujan identities and present new proofs of the finite versions of the Andrews-Gordon identity and the Bressoud identity. We also investigate the finite version of Garvan's generalizations of Dyson's rank and discover a new one-variable extension of the Andrews-Gordon identity.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106035"},"PeriodicalIF":0.9,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643721","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":"Characterizations of amorphic schemes and fusions of pairs","authors":"Edwin R. van Dam ,&nbsp;Jack H. Koolen ,&nbsp;Yanzhen Xiong","doi":"10.1016/j.jcta.2025.106045","DOIUrl":"10.1016/j.jcta.2025.106045","url":null,"abstract":"<div><div>An association scheme is called amorphic if every possible fusion of relations gives rise to a fusion scheme. We call a pair of relations fusing if fusing that pair gives rise to a fusion scheme. We define the fusing-relations graph on the set of relations, where a pair forms an edge if it fuses. We show that if the fusing-relations graph is connected but not a path, then the association scheme is amorphic. As a side result, we show that if an association scheme has at most one relation that is neither strongly regular of Latin square type nor strongly regular of negative Latin square type, then it is amorphic.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106045"},"PeriodicalIF":0.9,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143611456","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":"Distribution of maxima and minima statistics on alternating permutations, Springer numbers, and avoidance of flat POPs","authors":"Tian Han ,&nbsp;Sergey Kitaev ,&nbsp;Philip B. Zhang","doi":"10.1016/j.jcta.2025.106034","DOIUrl":"10.1016/j.jcta.2025.106034","url":null,"abstract":"<div><div>In this paper, we find distributions of the left-to-right maxima, right-to-left maxima, left-to-right minima and right-to-left-minima statistics on up-down and down-up permutations of even and odd lengths. We recover and generalize a result by Carlitz and Scoville, obtained in 1975, stating that the distribution of left-to-right maxima on down-up permutations of even length is given by <span><math><msup><mrow><mo>(</mo><mi>sec</mi><mo>⁡</mo><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup></math></span>. We also derive the joint distribution of the maxima (resp., minima) statistics, extending the scope of the respective results of Carlitz and Scoville, who obtain them in terms of certain systems of PDEs and recurrence relations. To accomplish this, we generalize a result of Kitaev and Remmel by deriving joint distributions involving non-maxima (resp., non-minima) statistics. Consequently, we refine classic enumeration results of André by introducing new <em>q</em>-analogues and <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-analogues for the number of alternating permutations.</div><div>Additionally, we verify Callan's conjecture (2012) that up-down permutations of even length fixed by reverse and complement are counted by the Springer numbers, thereby offering another combinatorial interpretation of these numbers. Furthermore, we propose two <em>q</em>-analogues and a <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-analogue of the Springer numbers. Lastly, we enumerate alternating permutations that avoid certain flat partially ordered patterns.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106034"},"PeriodicalIF":0.9,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511458","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":"Binary self-orthogonal codes which meet the Griesmer bound or have optimal minimum distances","authors":"Minjia Shi ,&nbsp;Shitao Li ,&nbsp;Tor Helleseth ,&nbsp;Jon-Lark Kim","doi":"10.1016/j.jcta.2025.106027","DOIUrl":"10.1016/j.jcta.2025.106027","url":null,"abstract":"<div><div>The purpose of this paper is two-fold. First, we characterize the existence of binary self-orthogonal codes meeting the Griesmer bound by employing the Solomon-Stiffler codes. As a result, we reduce a problem with an infinite number of cases to a finite number of cases. Second, we develop a general method to prove the nonexistence of some binary self-orthogonal codes by considering the residual code of a binary self-orthogonal code. Using such a characterization, we completely determine the exact value of <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>s</mi><mi>o</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mn>7</mn><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>s</mi><mi>o</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> denotes the largest minimum distance among all binary self-orthogonal <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>]</mo></math></span> codes.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106027"},"PeriodicalIF":0.9,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511263","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":"Flag transitive geometries with trialities and no dualities coming from Suzuki groups","authors":"Dimitri Leemans ,&nbsp;Klara Stokes ,&nbsp;Philippe Tranchida","doi":"10.1016/j.jcta.2025.106033","DOIUrl":"10.1016/j.jcta.2025.106033","url":null,"abstract":"<div><div>Recently, Leemans and Stokes constructed an infinite family of incidence geometries admitting trialities but no dualities from the groups <span><math><mi>P</mi><mi>S</mi><mi>L</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> (where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn><mi>n</mi></mrow></msup></math></span> with <em>p</em> a prime and <span><math><mi>n</mi><mo>&gt;</mo><mn>0</mn></math></span> a positive integer). Unfortunately, these geometries are not flag transitive. In this paper, we work with the Suzuki groups <span><math><mi>S</mi><mi>z</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>, where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>e</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> with <em>e</em> a positive integer and <span><math><mn>2</mn><mi>e</mi><mo>+</mo><mn>1</mn></math></span> is divisible by 3. For any odd integer <em>m</em> dividing <span><math><mi>q</mi><mo>−</mo><mn>1</mn></math></span>, <span><math><mi>q</mi><mo>+</mo><msqrt><mrow><mn>2</mn><mi>q</mi></mrow></msqrt><mo>+</mo><mn>1</mn></math></span> or <span><math><mi>q</mi><mo>−</mo><msqrt><mrow><mn>2</mn><mi>q</mi></mrow></msqrt><mo>+</mo><mn>1</mn></math></span> (i.e.: <em>m</em> is the order of some non-involutive element of <span><math><mi>S</mi><mi>z</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>), we construct geometries of type <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> that admit trialities but no dualities. We then prove that they are flag transitive when <span><math><mi>m</mi><mo>=</mo><mn>5</mn></math></span>, no matter the value of <em>q</em>. These geometries form the first infinite family of incidence geometries of rank 3 that are flag transitive and have trialities but no dualities. They are constructed using chamber systems and the trialities come from field automorphisms. These same geometries can also be considered as regular hypermaps with automorphism group <span><math><mi>S</mi><mi>z</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106033"},"PeriodicalIF":0.9,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510696","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}