Pub Date : 2025-08-08DOI: 10.1016/j.jcta.2025.106097
Yuxuan Li, Binzhou Xia, Sanming Zhou
A Cayley graph on the symmetric group <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>. For <mml:math altimg="si2.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>n</mml:mi></mml:math>, let <mml:math altimg="si267.svg"><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> be the set of <ce:italic>k</ce:italic>-cycles of <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> moving every point in <mml:math altimg="si4.svg"><mml:mo stretchy="false">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:math>. Recently, Siemons and Zalesski (2022) <ce:cross-ref ref>[26]</ce:cross-ref> posed a conjecture which is equivalent to saying that for any <mml:math altimg="si5.svg"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>5</mml:mn></mml:math> and <mml:math altimg="si2.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>n</mml:mi></mml:math> the nonnormal Cayley graph <mml:math altimg="si6.svg"><mml:mrow><mml:mi mathvariant="normal">Cay</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> on <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> with connection set <mml:math altimg="si267.svg"><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> has the Aldous property. Solving this conjecture, we prove that all these graphs have the Aldous property except when (i) <mml:math altimg="si7.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi
如果对称群Sn上的Cayley图的严格第二大特征值（即严格小于度的最大特征值）通过Sn的标准表示获得，则称其具有Aldous性质。对于1≤r<；k<n，设C（n,k;r）为Sn移动{1，…，r}中每一点的k个环的集合。最近，Siemons and Zalesski(2022)[26]提出了一个猜想，该猜想等价于对于任意n≥5且1≤r<；k<n，具有连接集C（n,k;r）的Sn上的非正态Cayley图Cay（Sn，C(n,k;r)）具有Aldous性质。通过求解这个猜想，我们证明了除(i) (n,k,r)=(6,5,1)或（ii） n为奇数，k =n−1，且1≤r<；n2外，所有图都具有Aldous性质。在此过程中，我们确定了Sn的所有不可约表示，这些表示可以实现Cay（Sn，C(n,n - 1;r)）的严格第二大特征值以及该图的最小特征值。
{"title":"The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups","authors":"Yuxuan Li, Binzhou Xia, Sanming Zhou","doi":"10.1016/j.jcta.2025.106097","DOIUrl":"https://doi.org/10.1016/j.jcta.2025.106097","url":null,"abstract":"A Cayley graph on the symmetric group <mml:math altimg=\"si1.svg\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of <mml:math altimg=\"si1.svg\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>. For <mml:math altimg=\"si2.svg\"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"><</mml:mo><mml:mi>n</mml:mi></mml:math>, let <mml:math altimg=\"si267.svg\"><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> be the set of <ce:italic>k</ce:italic>-cycles of <mml:math altimg=\"si1.svg\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> moving every point in <mml:math altimg=\"si4.svg\"><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:math>. Recently, Siemons and Zalesski (2022) <ce:cross-ref ref>[26]</ce:cross-ref> posed a conjecture which is equivalent to saying that for any <mml:math altimg=\"si5.svg\"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>5</mml:mn></mml:math> and <mml:math altimg=\"si2.svg\"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"><</mml:mo><mml:mi>n</mml:mi></mml:math> the nonnormal Cayley graph <mml:math altimg=\"si6.svg\"><mml:mrow><mml:mi mathvariant=\"normal\">Cay</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math> on <mml:math altimg=\"si1.svg\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> with connection set <mml:math altimg=\"si267.svg\"><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> has the Aldous property. Solving this conjecture, we prove that all these graphs have the Aldous property except when (i) <mml:math altimg=\"si7.svg\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"22 1","pages":"106097"},"PeriodicalIF":1.1,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144900111","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 : 2025-08-01Epub 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":"2025-08-01","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 : 2025-08-01Epub Date: 2025-04-03DOI: 10.1016/j.jcta.2025.106048
Shane Chern , Lin Jiu , Italo Simonelli
Given a positive integer n, consider a permutation of n 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 , the number of permutations needed for this process to end. In particular, we find explicit asymptotic expressions for the mean value , the variance , and higher central moments, and show that satisfies a central limit theorem.
{"title":"A central limit theorem for a card shuffling problem","authors":"Shane Chern , Lin Jiu , 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":"2025-08-01","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}
Pub Date : 2025-08-01Epub 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":"2025-08-01","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 : 2025-08-01Epub 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":"2025-08-01","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 : 2025-08-01Epub 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":"2025-08-01","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}
Pub Date : 2025-08-01Epub Date: 2025-02-27DOI: 10.1016/j.jcta.2025.106030
Peer Stelldinger
De Bruijn tori, or perfect maps, are two-dimensional periodic arrays of letters from a finite alphabet, where each possible pattern of shape appears exactly once in a single period. While the existence of certain de Bruijn tori, such as square tori with odd and even alphabet sizes, remains unresolved, sub-perfect maps are often sufficient in applications like positional coding. These maps capture a large number of patterns, with each appearing at most once. While previous methods for generating such sub-perfect maps cover only a fraction of the possible patterns, we present a construction method for generating almost perfect maps for arbitrary pattern shapes and arbitrary non-prime alphabet sizes, including the above mentioned square tori with odd as long that the alphabet size is non-prime. This is achieved through the introduction of de Bruijn rings, a minimal-height sub-perfect map and a formalization of the concept of families of almost perfect maps. The generated sub-perfect maps are easily decodable which makes them perfectly suitable for positional coding applications.
De Bruijn tori，或完美映射，是有限字母表中字母的二维周期性数组，其中每种可能的形状模式（m,n）在单个周期内恰好出现一次。虽然某些de Bruijn环面（例如奇数m=n∈{3,5,7}和偶数字母大小的平方环面）的存在性仍未得到解决，但在位置编码等应用中，次完美映射通常是足够的。这些地图捕获了大量的模式，每种模式最多出现一次。虽然以前生成这种次完美映射的方法只覆盖了可能模式的一小部分，但我们提出了一种生成任意模式形状和任意非素数字母表大小的几乎完美映射的构造方法，包括上面提到的奇数m=n∈{3,5,7}的方形环面，只要字母表大小是非素数。这是通过引入de Bruijn环，一个最小高度的次完美地图和几乎完美地图族概念的形式化来实现的。生成的次完美地图很容易解码，这使得它们非常适合位置编码应用。
{"title":"On de Bruijn rings and families of almost perfect maps","authors":"Peer Stelldinger","doi":"10.1016/j.jcta.2025.106030","DOIUrl":"10.1016/j.jcta.2025.106030","url":null,"abstract":"<div><div>De Bruijn tori, or perfect maps, are two-dimensional periodic arrays of letters from a finite alphabet, where each possible pattern of shape <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> appears exactly once in a single period. While the existence of certain de Bruijn tori, such as square tori with odd <span><math><mi>m</mi><mo>=</mo><mi>n</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>}</mo></math></span> and even alphabet sizes, remains unresolved, sub-perfect maps are often sufficient in applications like positional coding. These maps capture a large number of patterns, with each appearing at most once. While previous methods for generating such sub-perfect maps cover only a fraction of the possible patterns, we present a construction method for generating almost perfect maps for arbitrary pattern shapes and arbitrary non-prime alphabet sizes, including the above mentioned square tori with odd <span><math><mi>m</mi><mo>=</mo><mi>n</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>}</mo></math></span> as long that the alphabet size is non-prime. This is achieved through the introduction of de Bruijn rings, a minimal-height sub-perfect map and a formalization of the concept of families of almost perfect maps. The generated sub-perfect maps are easily decodable which makes them perfectly suitable for positional coding applications.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106030"},"PeriodicalIF":0.9,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510240","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 : 2025-08-01Epub Date: 2025-02-27DOI: 10.1016/j.jcta.2025.106028
Hau-Wen Huang
<div><div>The Clebsch–Gordan coefficients of <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism ♮ from the universal Hahn algebra <span><math><mi>H</mi></math></span> into <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. Let Ω denote a finite set of size <em>D</em> and <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></math></span> denote the power set of Ω. It is generally known that <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math></span> supports a <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-module. Let <em>k</em> denote an integer with <span><math><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>D</mi></math></span> and fix a <em>k</em>-element subset <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> of Ω. By identifying <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math></span> with <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi><mo>∖</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msup><mo>⊗</mo><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msup></math></span> this induces a <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-module structure on <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math></span> denoted by <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span>. Pulling back via ♮ the <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-module <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> forms an <span><math><mi>H</mi></math></span>-module. When <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>D</mi><mo>−</mo><mn>1
{"title":"An imperceptible connection between the Clebsch–Gordan coefficients of Uq(sl2) and the Terwilliger algebras of Grassmann graphs","authors":"Hau-Wen Huang","doi":"10.1016/j.jcta.2025.106028","DOIUrl":"10.1016/j.jcta.2025.106028","url":null,"abstract":"<div><div>The Clebsch–Gordan coefficients of <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism ♮ from the universal Hahn algebra <span><math><mi>H</mi></math></span> into <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. Let Ω denote a finite set of size <em>D</em> and <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></math></span> denote the power set of Ω. It is generally known that <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math></span> supports a <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-module. Let <em>k</em> denote an integer with <span><math><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>D</mi></math></span> and fix a <em>k</em>-element subset <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> of Ω. By identifying <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math></span> with <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi><mo>∖</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msup><mo>⊗</mo><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></mrow></msup></math></span> this induces a <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-module structure on <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math></span> denoted by <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span>. Pulling back via ♮ the <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>⊗</mo><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-module <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> forms an <span><math><mi>H</mi></math></span>-module. When <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>D</mi><mo>−</mo><mn>1","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106028"},"PeriodicalIF":0.9,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510465","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 : 2025-07-11DOI: 10.1016/j.jcta.2025.106093
Shi-Mei Ma , Hao Qi , Jean Yeh , Yeong-Nan Yeh
In the context of Stirling polynomials, Gessel and Stanley introduced Stirling permutations, which have attracted extensive attention over the past decades. Recently, we introduced Stirling permutation codes and provided numerous equidistribution results as applications. The purpose of the present work is to further analyze Stirling permutation codes. First, we derive an expansion formula expressing the joint distribution of the types A and B descent statistics over the hyperoctahedral group, and we also find an interlacing property involving the zeros of its coefficient polynomials. Next, we prove a strong connection between signed permutations in the hyperoctahedral group and Stirling permutations. We also study unified generalizations of the trivariate second-order Eulerian and ascent-plateau polynomials. Using Stirling permutation codes, we provide expansion formulas for eight-variable and seventeen-variable polynomials, which imply several e-positive expansions and clarify the connection among several statistics. Our results generalize the results of Bóna, Chen-Fu, Dumont, Haglund-Visontai, Janson and Petersen.
{"title":"Stirling permutation codes. II","authors":"Shi-Mei Ma , Hao Qi , Jean Yeh , Yeong-Nan Yeh","doi":"10.1016/j.jcta.2025.106093","DOIUrl":"10.1016/j.jcta.2025.106093","url":null,"abstract":"<div><div>In the context of Stirling polynomials, Gessel and Stanley introduced Stirling permutations, which have attracted extensive attention over the past decades. Recently, we introduced Stirling permutation codes and provided numerous equidistribution results as applications. The purpose of the present work is to further analyze Stirling permutation codes. First, we derive an expansion formula expressing the joint distribution of the types <em>A</em> and <em>B</em> descent statistics over the hyperoctahedral group, and we also find an interlacing property involving the zeros of its coefficient polynomials. Next, we prove a strong connection between signed permutations in the hyperoctahedral group and Stirling permutations. We also study unified generalizations of the trivariate second-order Eulerian and ascent-plateau polynomials. Using Stirling permutation codes, we provide expansion formulas for eight-variable and seventeen-variable polynomials, which imply several <em>e</em>-positive expansions and clarify the connection among several statistics. Our results generalize the results of Bóna, Chen-Fu, Dumont, Haglund-Visontai, Janson and Petersen.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106093"},"PeriodicalIF":0.9,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596379","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 : 2025-07-01Epub Date: 2025-01-20DOI: 10.1016/j.jcta.2025.106010
Zhicong Lin , Jun Ma
By considering the parity of the degrees and levels of nodes in increasing trees, a new combinatorial interpretation for the coefficients of the Taylor expansions of the Jacobi elliptic functions is found. As one application of this new interpretation, a conjecture of Ma–Mansour–Wang–Yeh is solved. Unifying the concepts of increasing trees and plane trees, Lin–Ma–Ma–Zhou introduced weakly increasing trees on a multiset. A symmetry joint distribution of “even-degree nodes on odd levels” and “odd-degree nodes” on weakly increasing trees is found, extending the Schett polynomials, a generalization of the Jacobi elliptic functions introduced by Schett, to multisets. A combinatorial proof and an algebraic proof of this symmetry are provided, as well as several relevant interesting consequences. Moreover, via introducing a group action on trees, we prove the partial γ-positivity of the multiset Schett polynomials, a result which implies both the symmetry and the unimodality of these polynomials.
{"title":"A symmetry on weakly increasing trees and multiset Schett polynomials","authors":"Zhicong Lin , Jun Ma","doi":"10.1016/j.jcta.2025.106010","DOIUrl":"10.1016/j.jcta.2025.106010","url":null,"abstract":"<div><div>By considering the parity of the degrees and levels of nodes in increasing trees, a new combinatorial interpretation for the coefficients of the Taylor expansions of the Jacobi elliptic functions is found. As one application of this new interpretation, a conjecture of Ma–Mansour–Wang–Yeh is solved. Unifying the concepts of increasing trees and plane trees, Lin–Ma–Ma–Zhou introduced weakly increasing trees on a multiset. A symmetry joint distribution of “even-degree nodes on odd levels” and “odd-degree nodes” on weakly increasing trees is found, extending the Schett polynomials, a generalization of the Jacobi elliptic functions introduced by Schett, to multisets. A combinatorial proof and an algebraic proof of this symmetry are provided, as well as several relevant interesting consequences. Moreover, via introducing a group action on trees, we prove the partial <em>γ</em>-positivity of the multiset Schett polynomials, a result which implies both the symmetry and the unimodality of these polynomials.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106010"},"PeriodicalIF":0.9,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143103357","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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