Journal of Combinatorial Theory Series A最新文献

英文中文

Regular ovoids and Cameron-Liebler sets of generators in polar spaces

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-25 DOI: 10.1016/j.jcta.2025.106029

Maarten De Boeck , Jozefien D'haeseleer , Morgan Rodgers

Cameron-Liebler sets of generators in polar spaces were introduced a few years ago as natural generalisations of the Cameron-Liebler sets of subspaces in projective spaces. In this article we present the first two constructions of non-trivial Cameron-Liebler sets of generators in polar spaces. Also regular m-ovoids of k-spaces are introduced as a generalization of m-ovoids of polar spaces. They are used in one of the aforementioned constructions of Cameron-Liebler sets.

极空间中的卡梅隆-利伯勒生成器集是几年前作为投影空间中子空间的卡梅隆-利伯勒集的自然广义而提出的。在这篇文章中，我们首次提出了极空间中非难卡梅隆-利伯勒生成器集的两个构造。此外，还介绍了 k 空间的正则 m-ovoids 作为极空间 m-ovoids 的广义。它们被用于上述卡梅隆-利伯勒集合的一个构造中。

引用次数: 0

Truncated forms of MacMahon's q-series

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-24 DOI: 10.1016/j.jcta.2025.106020

Mircea Merca

<div><div>In 1920, Percy Alexander MacMahon defined the partition generating functions<math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>:</mo><mo>=</mo><munder><mo>∑</mo><mrow><mn>0</mn><mo><</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></munder><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup></mrow><mrow><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>⋯</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math> and<math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>:</mo><mo>=</mo><munder><mo>∑</mo><mrow><mn>0</mn><mo><</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></munder><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>−</mo><mi>k</mi></mrow></msup></mrow><mrow><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>⋯</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math> which have since played an important rol

1920 年，珀西-亚历山大-麦克马洪定义了分区生成函数 Ak(q)：=∑0<n1<n2<⋯<nkqn1+n2+⋯+nk(1-qn1)2(1-qn2)2⋯(1-qnk)2 和 Ck(q)：=∑0<n1<n2<⋯<nkq2n1+2n2+⋯+2nk-k(1-q2n1-1)2(1-q2n2-1)2⋯(1-q2nk-1)2，它们在组合数学中发挥了重要作用。对于每一个非负整数 k，乔治-安德鲁斯（George E. Andrews）和西蒙-罗斯（Simon C. F. Rose）证明了 Ak(q)可以用分区的生成函数来表示，其中每一部分可以用三种不同颜色中的一种来着色，而 Ck(q)可以用过分区的生成函数来表示。最近，对于每个非负整数 k，Ken Ono 和 Ajit Singh 证明了 Ak(q)、Ak+1(q)、Ak+2(q)......给出了每个部分可以用三种不同颜色中的一种着色的 n 的分区数的生成函数，而 Ck(q)、Ck+1(q)、Ck+2(q)......给出了 n 的过度分区数的生成函数。本文还介绍了一些悬而未决的问题。

引用次数: 0

Simple geometric mitosis

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-24 DOI: 10.1016/j.jcta.2025.106022

Valentina Kiritchenko

We construct simple geometric operations on faces of the Cayley sum of two polytopes. These operations can be thought of as convex geometric counterparts of divided difference operators in Schubert calculus. We show that these operations give a uniform construction of Knutson–Miller mitosis in the type A and Fujita mitosis in the type C on Kogan faces of Gelfand–Zetlin polytopes.

引用次数: 0

On common energies and sumsets

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-24 DOI: 10.1016/j.jcta.2025.106026

Shkredov I.D.

We obtain a polynomial criterion for a set to have a small doubling in terms of the common energy of its subsets.

我们得到了一个多项式标准，即一个集合在其子集合的公共能量方面有一个小的加倍。

引用次数: 0

Structure of Terwilliger algebras of quasi-thin association schemes

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-02-24 DOI: 10.1016/j.jcta.2025.106024

Zhenxian Chen , Changchang Xi

We show that the Terwilliger algebra of a quasi-thin association scheme over a field is always a quasi-hereditary cellular algebra in the sense of Cline-Parshall-Scott and of Graham-Lehrer, respectively, and that the basic algebra of the Terwilliger algebra is the dual extension of a star with all arrows pointing to its center if the field has characteristic 2. Thus many homological and representation-theoretic properties of these Terwilliger algebras can be determined completely. For example, the Nakayama conjecture holds true for Terwilliger algebras of quasi-thin association schemes.

引用次数: 0

A symmetry on weakly increasing trees and multiset Schett polynomials

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-01-20 DOI: 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.

引用次数: 0

On recursive constructions for 2-designs over finite fields

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-01-20 DOI: 10.1016/j.jcta.2025.106006

Xiaoran Wang, Junling Zhou

This paper concentrates on recursive constructions for 2-designs over finite fields. In 1998, Itoh presented a powerful recursive construction: for certain index λ, if there exists a Singer cycle invariant 2-

{(l, 3, λ)}_{q}

design, then there also exists an SL

(m, q^{l})

invariant 2-

{(m l, 3, λ)}_{q}

design for all integers

m \geq 3

. We investigate the

GL (m, q^{l})

-incidence matrix between 2-subspaces and k-subspaces of

GF {(q)}^{m l}

with

m \geq 2

and

k \geq 3

in this work. As a generalization of Itoh's construction, the important case of

m = 2

is supplemented and a doubling construction is established for 2-

{(l, 3, λ)}_{q}

designs over finite fields. As a further generalization, a product construction is developed for q-analogs of group divisible designs (q-GDDs). For general block dimension

k \geq 3

, several new infinite families of q-GDDs are constructed. As applications, plenty of new infinite families of 2-designs over finite fields are constructed.

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引用次数: 0

On a conjecture concerning the r-Euler-Mahonian statistic on permutations

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-01-17 DOI: 10.1016/j.jcta.2025.106008

Kaimei Huang , Zhicong Lin , Sherry H.F. Yan

A pair

(s t_{1}, s t_{2})

of permutation statistics is said to be r-Euler-Mahonian if

(s t_{1}, s t_{2})

and (rdes, rmaj) are equidistributed over the set

S_{n}

of all permutations of

{1, 2, \dots, n}

, where rdes denotes the r-descent number and rmaj denotes the r-major index introduced by Rawlings. The main objective of this paper is to prove that

({exc}_{r}, {den}_{r})

and (rdes, rmaj) are equidistributed over

S_{n}

, thereby confirming a recent conjecture posed by Liu. When

r = 1

, the result recovers the equidistribution of

(des, maj)

and

(exc, den)

, which was first conjectured by Denert and proved by Foata and Zeilberger.

引用次数: 0

There are no good infinite families of toric codes

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-01-17 DOI: 10.1016/j.jcta.2025.106009

Jason P. Bell, Sean Monahan, Matthew Satriano, Karen Situ, Zheng Xie

Soprunov and Soprunova posed a question on the existence of infinite families of toric codes that are “good” in a precise sense. We prove that such good families do not exist by proving a more general Szemerédi-type result: for all

c \in (0, 1]

and all positive integers N, subsets of density at least c in

{0, 1, \dots, N - 1}^{n}

contain hypercubes of arbitrarily large dimension as n grows.

引用次数: 0

Unique representations of integers by linear forms

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-01-17 DOI: 10.1016/j.jcta.2025.106007

Sándor Z. Kiss , Csaba Sándor

Let

k \geq 2

be an integer and let A be a set of nonnegative integers. For a k-tuple of positive integers

\underline{λ} = (λ_{1}, \dots, λ_{k})

with

1 \leq λ_{1} < λ_{2} < \dots < λ_{k}

, we define the additive representation function

R_{A, \underline{λ}} (n) = | {(a_{1}, \dots, a_{k}) \in A^{k} : λ_{1} a_{1} + \dots + λ_{k} a_{k} = n} |

. For

k = 2

, Moser constructed a set A of nonnegative integers such that

R_{A, \underline{λ}} (n) = 1

holds for every nonnegative integer n. In this paper we characterize all the k-tuples

\underline{λ}

and the sets A of nonnegative integers with

R_{A, \underline{λ}} (n) = 1

for every integer

n \geq 0

引用次数: 0

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