Journal of Combinatorial Theory Series A最新文献

英文中文

Eulerian-type polynomials over Stirling permutations and box sorting algorithm 斯特林排列上的欧拉型多项式和盒排序算法

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-05-01 Epub Date: 2025-11-28 DOI: 10.1016/j.jcta.2025.106132

Shi-Mei Ma , Jun-Ying Liu , Jean Yeh , Yeong-Nan Yeh

It is well known that ascents, descents and plateaux are equidistributed over the set of classical Stirling permutations. Their common enumerative polynomials are the second-order Eulerian polynomials, which have been extensively studied by many researchers. This paper is divided into three parts. The first part gives a convolution formula for the second-order Eulerian polynomials, which simplifies a result of Gessel. As an application, a determinantal expression for the second-order Eulerian polynomial is obtained. We then investigate a convolution formula of the trivariate second-order Eulerian polynomials. Among other things, by introducing three new statistics: proper ascent-plateau, improper ascent-plateau and trace, we discover that a six-variable enumerative polynomial over restricted Stirling permutations equals a six-variable Eulerian-type polynomial over signed permutations. By special parametrizations, we make use of Stirling permutations to give a unified interpretation of the

(p, q)

-Eulerian polynomials and derangement polynomials of types A and B. The third part presents a box sorting algorithm which leads to a bijection between the terms in the expansion of

{(c D)}^{n} c

and ordered weak set partitions, where c is a smooth function in the indeterminate x and D is the derivative with respect to x. Using a map from ordered weak set partitions to standard Young tableaux, we find an expansion of

{(c D)}^{n} c

in terms of standard Young tableaux. Combining this with context-free grammars, we provide three new interpretations of the second-order Eulerian polynomials.

众所周知，在经典斯特林排列集合上，上升、下降和高原是均匀分布的。其常见的枚举多项式是二阶欧拉多项式，已被许多研究者广泛研究。本文共分为三个部分。第一部分给出了二阶欧拉多项式的卷积公式，简化了Gessel的结果。作为应用，得到了二阶欧拉多项式的行列式。然后我们研究了三元二阶欧拉多项式的卷积公式。此外，通过引入三个新的统计量：适当的上升-高原、不适当的上升-高原和迹，我们发现限制斯特林置换上的六变量枚举多项式等于符号置换上的六变量欧拉型多项式。通过特殊的参数化,我们使用斯特灵的排列给一个统一的解释(p, q)欧拉多项式多项式和错乱的类型a和b,第三部分提出了一个盒子排序算法导致条款之间的双射的扩张(cD)数控,命令弱设置分区,c是一个光滑函数的不定x和D是导数x。使用标准命令弱设置分区地图年轻的场景,我们发现了（cD）nc在标准杨氏表的展开式。结合上下文无关语法，我们提供了三种二阶欧拉多项式的新解释。

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引用次数: 0

Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation 非交换色拟对称函数，Macdonald多项式，和Yang-Baxter方程

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-05-01 Epub Date: 2025-11-21 DOI: 10.1016/j.jcta.2025.106130

Jean-Christophe Novelli, Jean-Yves Thibon

As shown in our paper (Novelli and Thibon, 2021 [20]), the chromatic quasi-symmetric function of Shareshian-Wachs can be lifted to WQSym, the algebra of quasi-symmetric functions in noncommuting variables. We investigate here its behavior with respect to classical transformations of alphabets and propose a noncommutative analogue of Macdonald polynomials compatible with a noncommutative version of the Haglund-Wilson formula. We also introduce a multi-t version of these noncommutative analogues. For rectangular partitions, their commutative images at

q = 0

appear to coincide with the multi-t Hall-Littlewood functions introduced in (Lascoux et al., 1995 [14]). This leads us to conjecture that for rectangular partitions, multi-t Macdonald polynomials are obtained as equivariant traces of certain Yang-Baxter elements of Hecke algebras. We also conjecture that all (ordinary) Macdonald polynomials can be obtained in this way. We conclude with some remarks relating various aspects of quasi-symmetric chromatic functions to calculations in Hecke algebras. In particular, we show that all modular relations are given by the product formula of the Kazhdan-Lusztig basis.

如我们的论文（Novelli and Thibon, 2021[20]）所示，Shareshian-Wachs的色拟对称函数可以提升到WQSym，即非交换变量中拟对称函数的代数。我们在这里研究了它在经典字母变换中的行为，并提出了一个与Haglund-Wilson公式的非交换版本兼容的Macdonald多项式的非交换模拟。我们还介绍了这些非交换类似物的多t版本。对于矩形分区，它们在q=0处的交换像似乎与（Lascoux et al., 1995[14]）中引入的多t Hall-Littlewood函数一致。这使我们推测，对于矩形分区，多重t Macdonald多项式可以作为Hecke代数的某些Yang-Baxter元素的等变迹得到。我们还推测所有（普通）麦克唐纳多项式都可以用这种方法得到。最后，我们对拟对称色函数的各个方面与Hecke代数中的计算作了一些评述。特别地，我们证明了所有模关系都是由Kazhdan-Lusztig基的乘积公式给出的。

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引用次数: 0

On cover-free families of finite vector spaces 有限向量空间的无复族

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-05-01 Epub Date: 2025-11-21 DOI: 10.1016/j.jcta.2025.106131

Yunjing Shan, Junling Zhou

There is a large literature on cover-free families of finite sets, because of their many applications in combinatorial group testing, cryptography and communications. This work studies the generalization of cover-free families from sets to finite vector spaces. Let V be an n-dimensional vector space over the finite field

F_{q}

and let

{[\frac{V}{k}]}_{q}

denote the family of all k-dimensional subspaces of V. A family

F \subseteq {[\frac{V}{k}]}_{q}

is called cover-free if there are no three distinct subspaces

F_{0}, F_{1}, F_{2} \in F

such that

F_{0} \leq (F_{0} \cap F_{1}) + (F_{0} \cap F_{2})

. A family

H \subseteq {[\frac{V}{k}]}_{q}

is called a q-Steiner system

S_{q} (t, k, n)

if for every

T \in {[\frac{V}{t}]}_{q}

, there is exactly one

H \in H

such that

T \leq H

. In this paper we investigate cover-free families in the vector space V. Firstly, we determine the maximum size of a cover-free family in

{[\frac{V}{k}]}_{q}

. Secondly, we characterize the structures of all maximum cover-free families which are closely related to q-Steiner systems.

由于有限集的无复族在组合群测试、密码学和通信等领域的广泛应用，有大量的文献研究有限集的无复族。本文研究了无复族从集合到有限向量空间的推广。设V是有限域Fq上的一个n维向量空间，设[Vk]q表示V的所有k维子空间的族。如果不存在三个不同的子空间F0、F1、F2∈F使F0≤(F0∩F1)+(F0∩F2)，则称族F≠Vk。如果对每一个t∈[Vt]q，有一个H∈H使t≤H，则称为q-施泰纳体系Sq（t,k,n）。本文研究了向量空间v中的无覆盖族，首先确定了[Vk]q中无覆盖族的最大大小。其次，我们刻画了与q-Steiner系统密切相关的所有最大无覆盖族的结构。

引用次数: 0

Computing the k-binomial complexity of generalized Thue–Morse words 计算广义Thue-Morse词的k-二项复杂度

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-05-01 Epub Date: 2025-12-15 DOI: 10.1016/j.jcta.2025.106152

M. Golafshan , M. Rigo , M.A. Whiteland

Two finite words are k-binomially equivalent if each subword (i.e., subsequence) of length at most k occurs the same number of times in both words. The k-binomial complexity of an infinite word is a function that maps the integer

n ⩾ 0

to the number of k-binomial equivalence classes represented by its factors of length n.

The Thue–Morse (TM) word and its generalization to larger alphabets are ubiquitous in mathematics due to their rich combinatorial properties. This work addresses the k-binomial complexities of generalized TM words. Prior research by Lejeune, Leroy, and Rigo determined the k-binomial complexities of the 2-letter TM word. For larger alphabets, work by Lü, Chen, Wen, and Wu determined the 2-binomial complexity for m-letter TM words, for arbitrary m, but the exact behavior for

k ⩾ 3

remained unresolved. They conjectured that the k-binomial complexity function of the m-letter TM word is eventually periodic with period

m^{k}

We resolve the conjecture positively by deriving explicit formulae for the k-binomial complexity functions for any generalized TM word. We do this by characterizing k-binomial equivalence among factors of generalized TM words. This comprehensive analysis not only solves the open conjecture, but also develops tools such as abelian Rauzy graphs.

如果长度最多为k的每个子词（即子序列）在两个词中出现相同次数，则两个有限词是k二项等价的。无限单词的k-二项复杂度是一个函数，它将整数n小于0映射到由长度n的因子表示的k-二项等价类的数量。

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引用次数: 0

Two results on set families: Sturdiness and intersection 集族的两个结果：坚固性和交性

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-05-01 Epub Date: 2025-12-05 DOI: 10.1016/j.jcta.2025.106153

Yongjiang Wu , Zhiyi Liu , Lihua Feng , Yongtao Li

This paper resolves two open problems in extremal set theory. For a family

F \subseteq 2^{[n]}

and

i, j \in [n]

, we denote

F (i, \bar{j}) = {F ﹨ {i} : F \in F, F \cap {i, j} = {i}}

. The sturdiness

β (F)

is defined as the minimum

| F (i, \bar{j}) |

over all

i \neq j

. A family

F

is called an IU-family if it satisfies the intersection constraint:

F \cap F^{'} \neq \emptyset

for all

F, F^{'} \in F

, as well as the union constraint:

F \cup F^{'} \neq [n]

for all

F, F^{'} \in F

. The well-known IU-Theorem states that every IU-family

F \subseteq 2^{[n]}

has size at most

2^{n - 2}

. In this paper, we prove that if

F \subseteq 2^{[n]}

is an IU-family, then

β (F) \leq 2^{n - 4}

. This confirms a recent conjecture proposed by Frankl and Wang.

As the second result, we establish a tight upper bound on the sum of sizes of cross t-intersecting separated families. Our result not only extends a previous theorem of Frankl, Liu, Wang and Yang on separated families, but also provides explicit counterexamples to an open problem proposed by them, thereby settling their problem in the negative.

本文解决了极值集理论中的两个开放问题。对于一个家庭F⊆2 [n]和我,j∈[n],我们表示F (i, j¯)={{我}:F∈F, F∩{i, j} ={我}}。坚固度β(F)定义为最小|F（i,j¯）|除以所有i≠j。如果一族F满足相交约束：F∩F′≠∅对于所有F，F′∈F，并且满足并约束：F∩F′≠[n]对于所有F，F′∈F。著名的u -定理认为，每一个u -族F≤2n−2。在本文中，我们证明了如果F≤2 [n]是一个u族，则β(F)≤2n−4。这证实了Frankl和Wang最近提出的一个猜想。作为第二个结果，我们建立了交叉t相交离散族大小总和的紧上界。我们的结果不仅推广了Frankl、Liu、Wang和Yang先前关于离散家庭的定理，而且为他们提出的一个开放问题提供了明确的反例，从而否定了他们的问题。

引用次数: 0

Partitions with Durfee triangles of fixed size 具有固定大小的Durfee三角形的分区

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-05-01 Epub Date: 2026-01-12 DOI: 10.1016/j.jcta.2026.106158

N. Guru Sharan , Armin Straub

A well-studied statistic of an integer partition is the size of its Durfee square. In particular, the number

D_{k} (n)

of partitions of n with Durfee square of fixed size k has a well-known simple rational generating function. We study the number

R_{k} (n)

of partitions of n with Durfee triangle of size k (the largest subpartition with parts

1, 2, \dots, k

). We determine the corresponding generating functions which are rational functions of a similar form. Moreover, we explicitly determine the leading asymptotic of

R_{k} (n)

, as

n \to \infty

整数分区的一个被充分研究的统计数据是它的Durfee平方的大小。特别是，具有固定大小k的Durfee平方的n分区的个数Dk(n)具有众所周知的简单有理生成函数。我们研究了大小为k的Durfee三角形（包含1,2，…，k部分的最大子划分）的n分区的个数Rk(n)。我们确定了相应的生成函数，它们是形式相似的有理函数。此外，我们明确地确定了Rk(n)的前渐近，即n→∞。

引用次数: 0

Superport networks 超级网络

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-05-01 Epub Date: 2025-12-01 DOI: 10.1016/j.jcta.2025.106134

P. Pylyavskyy , S. Shirokovskikh , M. Skopenkov

We study multiport networks, common in electrical engineering. They have boundary conditions different from electrical networks: the boundary vertices are split into pairs and the sum of the incoming currents is set to be zero in each pair. If one sets the voltage difference for each pair, then the incoming currents are uniquely determined. We generalize Kirchhoff's matrix-tree theorem to this setup. Different forests now contribute with different signs, making the proof subtle. In particular, we use the formula for the response matrix minors by R. Kenyon–D. Wilson, determinantal identities, and combinatorial bijections. We introduce superport networks, generalizing both ordinary networks and multiport ones.

我们研究电气工程中常见的多端口网络。它们的边界条件不同于电网络：边界顶点被分成对，每对输入电流的总和被设置为零。如果为每一对设置电压差，那么输入电流是唯一确定的。我们将基尔霍夫矩阵树定理推广到这种情况。不同的森林现在有不同的迹象，使证据变得微妙。特别地，我们使用了R. Kenyon-D的响应矩阵子式。Wilson，行列式恒等式和组合对偶。我们介绍了超级网络，对普通网络和多端口网络进行了推广。

引用次数: 0

Improved asymptotics for moments of reciprocal sums for partitions into distinct parts 分割成不同部分的互易和矩的改进渐近性

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-04-01 Epub Date: 2025-09-26 DOI: 10.1016/j.jcta.2025.106119

Kathrin Bringmann , Byungchan Kim , Eunmi Kim

In this paper we strongly improve asymptotics for

s_{1} (n)

(respectively

s_{2} (n)

) which sums reciprocals (respectively squares of reciprocals) of parts throughout all the partitions of n into distinct parts. The methods required are much more involved than in the case of usual partitions since the generating functions are not modular and also do not possess product expansions.

本文强烈改进了s1(n)（分别为s2(n)）的渐近性，该渐近性对n的所有划分为不同部分的部分的倒数（分别为倒数的平方）求和。由于生成函数不是模块化的，也不具有乘积展开，因此所需的方法比通常分区的情况要复杂得多。

引用次数: 0

Dimension identities, almost self-conjugate partitions, and BGG complexes for Hermitian symmetric pairs 厄密对称对的维恒等式、几乎自共轭分割和BGG复合体

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-04-01 Epub Date: 2025-09-25 DOI: 10.1016/j.jcta.2025.106118

William Q. Erickson, Markus Hunziker

An almost self-conjugate (ASC) partition has a Young diagram in which each arm along the diagonal is exactly one box longer than its corresponding leg. Classically, the ASC partitions and their conjugates appear in two of Littlewood's symmetric function identities. These identities can be viewed as Euler characteristics of BGG complexes of the trivial representation, for classical Hermitian symmetric pairs. In this paper, we consider partitions in which the arm–leg difference is an arbitrary constant m. By viewing these partitions as highest weights, we establish an infinite family of dimension identities between

{gl}_{n}

- and

{gl}_{n + m}

-modules. We then interpret this result in the context of blocks in parabolic category

O

: in particuar, we exhibit six infinite families of congruent blocks whose corresponding posets of highest weights consist of the partitions in question. These posets, in turn, lead to generalizations of the Littlewood identities and their corresponding BGG complexes. Our results in this paper shed light on the surprising combinatorics underlying the work of Enright and Willenbring (2004).

几乎自共轭（ASC）划分有一个杨图，其中沿对角线的每条臂正好比其相应的腿长一个盒子。经典地，ASC划分及其共轭出现在两个Littlewood对称函数恒等式中。这些恒等式可以看作是经典厄密对称对的平凡表示的BGG复合体的欧拉特征。在本文中，我们考虑臂腿差为任意常数m的分区。通过将这些分区视为最高权值，我们建立了gln-和gln+m模块之间的无限维恒等式。然后，我们在抛物线类别O中的块的背景下解释这个结果：特别是，我们展示了六个无限的同余块族，其相应的最大权重的偏置集由所讨论的分区组成。反过来，这些偏序集导致了Littlewood恒等式及其相应的BGG复合体的推广。我们在这篇论文中的结果揭示了Enright和Willenbring（2004）的工作背后令人惊讶的组合。

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引用次数: 0

Dissection of the quintuple product, with applications 五元积的解剖，及其应用

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-04-01 Epub Date: 2025-10-13 DOI: 10.1016/j.jcta.2025.106122

Tim Huber , James Mc Laughlin , Dongxi Ye

<div><div>This work considers the m-dissection (for <math><mi>m</mi><mo>≢</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math>) of the general quintuple product<math><mi>Q</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>q</mi><mo>/</mo><mi>z</mi><mo>,</mo><mi>q</mi><mo>;</mo><mi>q</mi><mo>)</mo></mrow><mrow><mo>∞</mo></mrow></msub><msub><mrow><mo>(</mo><mi>q</mi><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi>q</mi><mo>/</mo><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>;</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>∞</mo></mrow></msub><mo>.</mo></math> Multiple novel applications arise from this m-dissection. For example, we derive the general partition identity<math><msub><mrow><mi>D</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>m</mi><mi>n</mi><mo>+</mo><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>24</mn><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>6</mn></mrow></msup><msub><mrow><mi>b</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>,</mo><mspace></mspace><mtext> for all </mtext><mi>n</mi><mo>≥</mo><mn>0</mn><mo>,</mo></math> where <math><mi>m</mi><mo>≡</mo><mn>5</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>6</mn><mo>)</mo></math> is a square-free positive integer relatively prime to 6; <math><msub><mrow><mi>D</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> is defined, for S the set of positive integers containing no multiples of m, to be the number of partitions of n into an even number of distinct parts from S minus the number of partitions of n into an odd number of distinct parts from S; and <math><msub><mrow><mi>b</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> denotes the number of m-regular partitions of n. The dissections allow us to prove a conjecture of Hirschhorn concerning the <math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math>-dissection of <math><msub><mrow><mo>(</mo><mi>q</mi><mo>;</mo><mi>q</mi><mo>)</mo></mrow><mrow><mo>∞</mo></mrow></msub></math>, as well as determine the pattern of the sign changes of the coefficients <math><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math> of the infinite product<math><mfrac><mrow><msub><mrow><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mro

本文考虑一般五元积tq (z,q)=(z,q/z,q;q)∞(qz2,q/z2;q2)∞的m-剖分（对于m > 0(mod3)）。这种m型解剖产生了多种新的应用。例如，我们推导出一般划分恒等式ds (mn+(m2−1)/24)=(−1)(m+1)/6bm(n)，对于所有n≥0，其中m≡5（mod6）是一个相对素数为6的无平方正整数；定义DS(n)，对于S是不含m的正整数集合，等于n被划分为从S出发的偶数个可分离部分的个数减去n被划分为从S出发的奇数个可分离部分的个数；bm(n)表示n的m个正则分割的个数。这些分割证明了Hirschhorn关于(q;q)∞的2n-分割的一个猜想，并确定了无穷积(q2k−1;q2k−1)∞(qp;qp)∞2=∑n=0∞和qn，k≥1，p≥5a素数的系数an的符号变化规律。这涵盖了Bringmann等人最近的结果，对应于k=1和p=5的情况。

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