Let Γ be a Cayley graph on a finite group G, and let be the normalizer of (the right regular representation of G) in the full automorphism group of Γ. We say that Γ is a normal Cayley graph on G if , and that Γ is a normal edge-transitive Cayley graph on G if acts transitively on the edge set of Γ. 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 p is normal for each prime p. 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 G is either for an odd prime , or for , then there exists a connected normal edge-transitive but non-normal 8-valent Cayley graph of G.
{"title":"Normal edge-transitive Cayley graphs on non-abelian simple groups","authors":"Xing Zhang, Yan-Quan Feng, Fu-Gang Yin, 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}
Pub Date : 2025-03-18DOI: 10.1016/j.jcta.2025.106035
Heng Huat Chan , Song Heng Chan
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.
{"title":"Finite versions of the Andrews–Gordon identity and Bressoud's identity","authors":"Heng Huat Chan , 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}
Pub Date : 2025-03-13DOI: 10.1016/j.jcta.2025.106045
Edwin R. van Dam , Jack H. Koolen , Yanzhen Xiong
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.
{"title":"Characterizations of amorphic schemes and fusions of pairs","authors":"Edwin R. van Dam , Jack H. Koolen , 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}
Pub Date : 2025-02-28DOI: 10.1016/j.jcta.2025.106034
Tian Han , Sergey Kitaev , Philip B. Zhang
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 . 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 q-analogues and -analogues for the number of alternating permutations.
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 q-analogues and a -analogue of the Springer numbers. Lastly, we enumerate alternating permutations that avoid certain flat partially ordered patterns.
{"title":"Distribution of maxima and minima statistics on alternating permutations, Springer numbers, and avoidance of flat POPs","authors":"Tian Han , Sergey Kitaev , 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}
Pub Date : 2025-02-28DOI: 10.1016/j.jcta.2025.106027
Minjia Shi , Shitao Li , Tor Helleseth , Jon-Lark Kim
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 , where denotes the largest minimum distance among all binary self-orthogonal codes.
{"title":"Binary self-orthogonal codes which meet the Griesmer bound or have optimal minimum distances","authors":"Minjia Shi , Shitao Li , Tor Helleseth , 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}
Pub Date : 2025-02-27DOI: 10.1016/j.jcta.2025.106033
Dimitri Leemans , Klara Stokes , Philippe Tranchida
Recently, Leemans and Stokes constructed an infinite family of incidence geometries admitting trialities but no dualities from the groups (where with p a prime and a positive integer). Unfortunately, these geometries are not flag transitive. In this paper, we work with the Suzuki groups , where with e a positive integer and is divisible by 3. For any odd integer m dividing , or (i.e.: m is the order of some non-involutive element of ), we construct geometries of type that admit trialities but no dualities. We then prove that they are flag transitive when , no matter the value of q. 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 .
{"title":"Flag transitive geometries with trialities and no dualities coming from Suzuki groups","authors":"Dimitri Leemans , Klara Stokes , 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>></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}
Pub Date : 2025-02-27DOI: 10.1016/j.jcta.2025.106023
Martino Borello , Wolfgang Schmid , Martin Scotti
Intersecting codes are linear codes where every two nonzero codewords have non-trivially intersecting support. In this article we expand on the theory of this family of codes, by showing that nondegenerate intersecting codes correspond to sets of points (with multiplicities) in a projective space that are not contained in two hyperplanes. This correspondence allows the use of geometric arguments to demonstrate properties and provide constructions of intersecting codes. We improve on existing bounds on their length and provide explicit constructions of short intersecting codes. Finally, generalizing a link between coding theory and the theory of the Davenport constant (a combinatorial invariant of finite abelian groups), we provide new asymptotic bounds on the weighted 2-wise Davenport constant. These bounds then yield results on factorizations in rings of algebraic integers and related structures.
{"title":"The geometry of intersecting codes and applications to additive combinatorics and factorization theory","authors":"Martino Borello , Wolfgang Schmid , Martin Scotti","doi":"10.1016/j.jcta.2025.106023","DOIUrl":"10.1016/j.jcta.2025.106023","url":null,"abstract":"<div><div>Intersecting codes are linear codes where every two nonzero codewords have non-trivially intersecting support. In this article we expand on the theory of this family of codes, by showing that nondegenerate intersecting codes correspond to sets of points (with multiplicities) in a projective space that are not contained in two hyperplanes. This correspondence allows the use of geometric arguments to demonstrate properties and provide constructions of intersecting codes. We improve on existing bounds on their length and provide explicit constructions of short intersecting codes. Finally, generalizing a link between coding theory and the theory of the Davenport constant (a combinatorial invariant of finite abelian groups), we provide new asymptotic bounds on the weighted 2-wise Davenport constant. These bounds then yield results on factorizations in rings of algebraic integers and related structures.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106023"},"PeriodicalIF":0.9,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510238","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}
Pub Date : 2025-02-27DOI: 10.1016/j.jcta.2025.106021
Ming Liu, Houyi Yu
Separable elements in Weyl groups are generalizations of the well-known class of separable permutations in symmetric groups. Gaetz and Gao showed that for any pair of subsets of the symmetric group , the multiplication map is a splitting (i.e., a length-additive bijection) of if and only if X is the generalized quotient of Y and Y is a principal lower order ideal in the right weak order generated by a separable element. They conjectured this result can be extended to all finite Weyl groups. In this paper, we classify all separable and minimal non-separable signed permutations in terms of forbidden patterns and confirm the conjecture of Gaetz and Gao for Weyl groups of type B.
{"title":"Separable elements and splittings in Weyl groups of type B","authors":"Ming Liu, Houyi Yu","doi":"10.1016/j.jcta.2025.106021","DOIUrl":"10.1016/j.jcta.2025.106021","url":null,"abstract":"<div><div>Separable elements in Weyl groups are generalizations of the well-known class of separable permutations in symmetric groups. Gaetz and Gao showed that for any pair <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> of subsets of the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the multiplication map <span><math><mi>X</mi><mo>×</mo><mi>Y</mi><mo>→</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is a splitting (i.e., a length-additive bijection) of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> if and only if <em>X</em> is the generalized quotient of <em>Y</em> and <em>Y</em> is a principal lower order ideal in the right weak order generated by a separable element. They conjectured this result can be extended to all finite Weyl groups. In this paper, we classify all separable and minimal non-separable signed permutations in terms of forbidden patterns and confirm the conjecture of Gaetz and Gao for Weyl groups of type <em>B</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106021"},"PeriodicalIF":0.9,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510239","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}
Pub Date : 2025-02-27DOI: 10.1016/j.jcta.2025.106025
Eiichi Bannai , Hirotake Kurihara , Da Zhao , Yan Zhu
The classification problem of P- and Q-polynomial association schemes has been one of the central problems in algebraic combinatorics. Generalizing the concept of P- and Q-polynomial association schemes to multivariate cases, namely to consider higher rank P- and Q-polynomial association schemes, has been tried by some authors, but it seems that so far there were neither very well-established definitions nor results. Very recently, Bernard, Crampé, d'Andecy, Vinet, and Zaimi [4], defined bivariate P-polynomial association schemes, as well as bivariate Q-polynomial association schemes. In this paper, we study these concepts and propose a new modified definition concerning a general monomial order, which is more general and more natural and also easy to handle. We prove that there are many interesting families of examples of multivariate P- and/or Q-polynomial association schemes.
{"title":"Multivariate P- and/or Q-polynomial association schemes","authors":"Eiichi Bannai , Hirotake Kurihara , Da Zhao , Yan Zhu","doi":"10.1016/j.jcta.2025.106025","DOIUrl":"10.1016/j.jcta.2025.106025","url":null,"abstract":"<div><div>The classification problem of <em>P</em>- and <em>Q</em>-polynomial association schemes has been one of the central problems in algebraic combinatorics. Generalizing the concept of <em>P</em>- and <em>Q</em>-polynomial association schemes to multivariate cases, namely to consider higher rank <em>P</em>- and <em>Q</em>-polynomial association schemes, has been tried by some authors, but it seems that so far there were neither very well-established definitions nor results. Very recently, Bernard, Crampé, d'Andecy, Vinet, and Zaimi <span><span>[4]</span></span>, defined bivariate <em>P</em>-polynomial association schemes, as well as bivariate <em>Q</em>-polynomial association schemes. In this paper, we study these concepts and propose a new modified definition concerning a general monomial order, which is more general and more natural and also easy to handle. We prove that there are many interesting families of examples of multivariate <em>P</em>- and/or <em>Q</em>-polynomial association schemes.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106025"},"PeriodicalIF":0.9,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510697","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}
Pub Date : 2025-02-27DOI: 10.1016/j.jcta.2025.106032
Y.H. Chen, Thomas Y. He
Bressoud introduced the partition function , which counts the number of partitions with certain difference conditions. Bressoud posed a conjecture on the generating function for the partition function in multi-summation form. In this article, we introduce a bijection related to Bressoud's conjecture. As an application, we give the proof of a companion to the Göllnitz-Gordon identities.
{"title":"A bijection related to Bressoud's conjecture","authors":"Y.H. Chen, Thomas Y. He","doi":"10.1016/j.jcta.2025.106032","DOIUrl":"10.1016/j.jcta.2025.106032","url":null,"abstract":"<div><div>Bressoud introduced the partition function <span><math><mi>B</mi><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>;</mo><mi>η</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>;</mo><mi>n</mi><mo>)</mo></math></span>, which counts the number of partitions with certain difference conditions. Bressoud posed a conjecture on the generating function for the partition function <span><math><mi>B</mi><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>;</mo><mi>η</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo>;</mo><mi>n</mi><mo>)</mo></math></span> in multi-summation form. In this article, we introduce a bijection related to Bressoud's conjecture. As an application, we give the proof of a companion to the Göllnitz-Gordon identities.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106032"},"PeriodicalIF":0.9,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510241","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}
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