Journal of Combinatorial Theory Series A最新文献

英文中文

Distribution of maxima and minima statistics on alternating permutations, Springer numbers, and avoidance of flat POPs

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-28 DOI: 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

{(\sec (t))}^{q}

. 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

(p, q)

-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

(p, q)

-analogue of the Springer numbers. Lastly, we enumerate alternating permutations that avoid certain flat partially ordered patterns.

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引用次数: 0

Binary self-orthogonal codes which meet the Griesmer bound or have optimal minimum distances

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-28 DOI: 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

d_{s o} (n, 7)

, where

d_{s o} (n, k)

denotes the largest minimum distance among all binary self-orthogonal

[n, k]

codes.

引用次数: 0

Flag transitive geometries with trialities and no dualities coming from Suzuki groups

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-27 DOI: 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

P S L (2, q)

(where

q = p^{3 n}

with p a prime and

n > 0

a positive integer). Unfortunately, these geometries are not flag transitive. In this paper, we work with the Suzuki groups

S z (q)

, where

q = 2^{2 e + 1}

with e a positive integer and

2 e + 1

is divisible by 3. For any odd integer m dividing

q - 1

q + \sqrt{2 q} + 1

q - \sqrt{2 q} + 1

(i.e.: m is the order of some non-involutive element of

S z (q)

), we construct geometries of type

(m, m, m)

that admit trialities but no dualities. We then prove that they are flag transitive when

m = 5

, 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

S z (q)

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引用次数: 0

The geometry of intersecting codes and applications to additive combinatorics and factorization theory

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-27 DOI: 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.

引用次数: 0

Separable elements and splittings in Weyl groups of type B

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-27 DOI: 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

(X, Y)

of subsets of the symmetric group

S_{n}

, the multiplication map

X \times Y \to S_{n}

is a splitting (i.e., a length-additive bijection) of

S_{n}

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.

引用次数: 0

Multivariate P- and/or Q-polynomial association schemes

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-27 DOI: 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.

引用次数: 0

A bijection related to Bressoud's conjecture

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-27 DOI: 10.1016/j.jcta.2025.106032

Y.H. Chen, Thomas Y. He

Bressoud introduced the partition function

B (α_{1}, \dots, α_{λ}; η, k, r; n)

, which counts the number of partitions with certain difference conditions. Bressoud posed a conjecture on the generating function for the partition function

B (α_{1}, \dots, α_{λ}; η, k, r; n)

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.

引用次数: 0

On de Bruijn rings and families of almost perfect maps

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-27 DOI: 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

(m, n)

appears exactly once in a single period. While the existence of certain de Bruijn tori, such as square tori with odd

m = n \in {3, 5, 7}

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

m = n \in {3, 5, 7}

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.

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引用次数: 0

An imperceptible connection between the Clebsch–Gordan coefficients of Uq(sl2) and the Terwilliger algebras of Grassmann graphs

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-27 DOI: 10.1016/j.jcta.2025.106028

Hau-Wen Huang

<div><div>The Clebsch–Gordan coefficients of <math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism ♮ from the universal Hahn algebra <math><mi>H</mi></math> into <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>. Let Ω denote a finite set of size D and <math><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></math> denote the power set of Ω. It is generally known that <math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math> supports a <math><mi>U</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math>-module. Let k denote an integer with <math><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>D</mi></math> and fix a k-element subset <math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math> of Ω. By identifying <math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math> with <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> this induces a <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>-module structure on <math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi></mrow></msup></mrow></msup></math> denoted by <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>. Pulling back via ♮ the <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>-module <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> forms an <math><mi>H</mi></math>-module. When <math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>D</mi><mo>−</mo><mn>1

引用次数: 0

More on r-cross t-intersecting families for vector spaces

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-26 DOI: 10.1016/j.jcta.2025.106031

Tian Yao , Dehai Liu , Kaishun Wang

Let V be a finite dimensional vector space over a finite field. Suppose that

F_{1}

F_{2}

, …,

F_{r}

are r-cross t-intersecting families of k-subspaces of V. In this paper, we determine the extremal structure when

\prod_{i = 1}^{r} | F_{i} |

is maximum under the condition that

\dim (⋂_{F \in F_{i}} F) < t

for each i.

引用次数: 0

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