Journal of Combinatorial Theory Series A最新文献

英文中文

Mean and variance of the longest alternating subsequence in a random separable permutation 随机可分排列中最长交替子序列的均值和方差

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-12-31 DOI: 10.1016/j.jcta.2025.106157

Ross G. Pinsky

<div><div>A permutation is separable if it can be obtained from the singleton permutation by iterating direct sums and skew sums. Equivalently, it is separable if and only it avoids the patterns 2413 and 3142. Under the uniform probability on separable permutations of <math><mo>[</mo><mi>n</mi><mo>]</mo></math>, let the random variable <math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math> denote the length of the longest alternating subsequence. Also, let <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo><mo>,</mo><mo>−</mo></mrow></msubsup></math> denote the length of the longest alternating subsequence that begins with an ascent and ends with a descent, and define <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mo>,</mo><mo>+</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo><mo>,</mo><mo>+</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mo>,</mo><mo>−</mo></mrow></msubsup></math> similarly. By symmetry, the first two and the last two of these latter four random variables are equi-distributed. We prove that the expected value of any of these five random variables behaves asymptotically as <math><mo>(</mo><mn>2</mn><mo>−</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>)</mo><mi>n</mi><mo>≈</mo><mn>0.5858</mn><mspace></mspace><mi>n</mi></math>. We also obtain the more refined estimates that the expected value of <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo><mo>,</mo><mo>−</mo></mrow></msubsup></math> and of <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mo>,</mo><mo>+</mo></mrow></msubsup></math> is equal to <math><mo>(</mo><mn>2</mn><mo>−</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>)</mo><mi>n</mi><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>(</mo><mn>3</mn><mo>−</mo><mn>2</mn><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>)</mo><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> and that the expected value of <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo><mo>,</mo><mo>+</mo></mrow></msubsup></math> and of <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mo>,</mo><mo>−</mo></mrow></msubsup></math> is equal to <math><mo>(</mo><mn>2</mn><mo>−</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>)</mo><mi>n</mi><mo>+</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>(</mo><mn>3</mn><mo>−</mo><mn>2</mn><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>)</mo><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math>. Finally, we show that the variance of any of the four random variables <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>±</mo

一个排列是可分离的，如果它能从单态排列中通过迭代直接和和和得到。同样，当且仅当它避免了模式2413和3142时，它是可分离的。在可分离排列[n]的均匀概率下，设随机变量An表示最长交替子序列的长度。同样，设An+，−表示以上升开始，以下降结束的最长交替子序列的长度，并类似地定义An−，+,An+,+,An−,−。根据对称性，后四个随机变量的前两个和后两个是等分布的。我们证明了这五个随机变量的期望值的渐近性为（2−2）n≈0.5858n。我们还得到了An+，−和An−，+的期望值等于（2−2）n−14(3−22)+o(1)和An+，+和An−，−的期望值等于（2−2）n+34(3−22)+o(1)的更精确估计。最后，我们证明了任意四个随机变量的方差An±，±的渐近表现为16−1122n≈0.2218n。

引用次数: 0

Periods of strongly connected multivariate digraphs 强连通多元有向图的周期

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-12-22 DOI: 10.1016/j.jcta.2025.106156

Chengyang Qian, Yaokun Wu, Yinfeng Zhu

For a positive integer t, a t-variable digraph on a set K is defined as a map f from

K^{t}

2^{K}

. Note that an ordinary digraph is a 1-variable digraph. In 2017, Wu, Xu, and Zhu proposed the study of multivariate digraphs as a qualitative counterpart of going from Markov chains to higher-order Markov chains. A fundamental parameter of any strongly connected ordinary digraph is its period. This notion extends naturally to strongly connected t-variable digraphs. Let

PS (t)

denote the set of all possible periods of strongly connected t-variable digraphs, let

g (t)

be its Frobenius number (i.e., the largest nonnegative integer not belonging to

PS (t)

), and let

n (t)

be its Sylvester number (i.e., the number of positive integers outside of

PS (t)

). In this paper, we establish new estimates for

g (t)

and

n (t)

. We also show that

PS (t) \cap {1, 2, \dots, 4 t - 1}

equals

{1, 8}

when

t \in {3, 4}

and {1} when

t \geq 5

. Although this work originated in an effort to understand qualitative higher-order Markov chains, it turns out to be closely related to two other active areas of research, partitioning a discrete box into subboxes, and restricted universal partial cycles.

对于正整数t，集合K上的t变量有向图定义为从Kt到2K的映射。注意，普通有向图是一个单变量有向图。2017年，Wu、Xu和Zhu提出了多元有向图的研究，作为从马尔可夫链到高阶马尔可夫链的定性对应物。任何强连通普通有向图的一个基本参数是它的周期。这个概念自然地扩展到强连接的t变量有向图。设PS(t)表示强连通t变量有向图的所有可能周期的集合，设g(t)为它的Frobenius数（即不属于PS(t)的最大非负整数），设n(t)为它的Sylvester数（即不属于PS(t)的正整数的个数）。本文建立了g(t)和n(t)的新估计。我们还证明了当t∈{3,4}时，PS(t)∩{1,2，…，4t−1}={1,8}，当t≥5时，p (t)∩{1}。虽然这项工作起源于理解定性高阶马尔可夫链的努力，但事实证明，它与另外两个活跃的研究领域密切相关，将离散盒划分为子盒，以及限制泛环。

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引用次数: 0

On edge-primitive Cayley graphs 在边基Cayley图上

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-12-22 DOI: 10.1016/j.jcta.2025.106155

Zai Ping Lu, Yue Sun

A graph is said to be edge-primitive if its automorphism group acts primitively on the edge set. In this paper, we investigate finite edge-primitive Cayley graphs of valency no less than 2. An explicit classification of such graphs is obtained in the case where the graphs admit an almost simple edge-primitive automorphism group which contains a regular subgroup on the vertices. This implies that the only edge-primitive Cayley graphs of valency at least 2 defined over simple groups are cycles with prime length and complete graphs. In addition, we also classify those edge-primitive Cayley graphs which are either 2-arc-transitive or of square-free order.

如果图的自同构群基本作用于边集，则称图为边基图。本文研究了价不小于2的有限边基Cayley图。当图承认一个几乎简单的边原自同构群，且该群在顶点上包含正则子群时，得到了这类图的显式分类。这表明在单群上定义的价至少为2的边基Cayley图只有素数长度的环和完全图。此外，我们还对2-弧传递和无平方阶的边基Cayley图进行了分类。

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 : 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

Lambert series and double Lambert series 朗伯级数和二重朗伯级数

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-12-11 DOI: 10.1016/j.jcta.2025.106154

Tewodros Amdeberhan , George E. Andrews , Cristina Ballantine

We consider relationships between classical Lambert series, multiple Lambert series and classical q-series of the Rogers-Ramanujan type. We conclude with a contemplation on the Andrews-Dixit-Schultz-Yee conjecture.

研究Rogers-Ramanujan型的经典Lambert级数、多重Lambert级数与经典q级数之间的关系。最后，我们对Andrews-Dixit-Schultz-Yee猜想进行了思考。

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 : 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

Complete mappings stabilizing a subgroup and their parities 稳定子群及其对偶的完全映射

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-12-05 DOI: 10.1016/j.jcta.2025.106143

Shikang Yu , Tao Feng , Hengrui Liu

A complete mapping of a finite group G is a permutation

ϕ : G \to G

such that

x \mapsto x ϕ (x)

is also a permutation of G. The complete mapping ϕ is called odd (resp. even) when ϕ is an odd (resp. even) permutation. To address whether the permutation groups generated by all complete mappings of a finite group G are “large” and primitive, Bors and Wang (2023) initiated the investigation into which groups admit both even and odd complete mappings. It is conjectured that such groups exist if and only if their Sylow 2-subgroups are trivial or noncyclic, except for some groups of small order. This paper reduces the problem of determining whether an arbitrary finite group admits both even and odd complete mappings to examining (1) the existence of complete mappings that stabilize a Sylow 2-subgroup for every finite simple group of Lie type and every sporadic simple group in the set

{Ly, {Co}_{2}, B, Th, {Fi}_{22}, {Fi}_{23}, {Fi}_{24}^{'}, Ru, O^{'} N, {Co}_{1}, {Co}_{3}, HN, M}

, and (2) the existence of complete mappings that stabilize a subgroup of order

8 p_{0}^{t_{0}}

(for some odd prime

p_{0}

and

t_{0} ⩾ 1

) for every Ree group

{}^{2}G_{2} (3^{2 n + 1})

with

n ⩾ 1

有限群G的完全映射是一个φ:G→G的置换，使得x∑xφ (x)也是G的置换。偶数)，当φ是奇数时。甚至)排列。为了解决由有限群G的所有完备映射生成的置换群是否“大”且原始的问题，Bors和Wang（2023）开始研究哪些群同时存在奇偶完备映射。我们推测，除了一些小阶群外，这样的群当且仅当它们的Sylow 2-子群是平凡的或非循环的。本文将确定任意有限群是否存在奇偶完全映射的问题简化为检验(1)对于集合{Ly，Co2，B,Th,Fi22,Fi23,Fi24 ',Ru,O ' n,Co1,Co3，HN，M}中的每一个Lie型有限简单群和每一个偶然性简单群，是否存在稳定Sylow 2-子群的完全映射；以及(2)对于具有n大于或等于1的每个Ree组G22（32n+1）稳定8p0阶子群（对于一些奇数素数p0和t0小于或等于1）的完整映射的存在。

{"title":"Complete mappings stabilizing a subgroup and their parities","authors":"Shikang Yu , Tao Feng , Hengrui Liu","doi":"10.1016/j.jcta.2025.106143","DOIUrl":"10.1016/j.jcta.2025.106143","url":null,"abstract":"<div><div>A complete mapping of a finite group G is a permutation <math><mi>ϕ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi></math> such that <math><mi>x</mi><mo>↦</mo><mi>x</mi><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is also a permutation of G. The complete mapping ϕ is called odd (resp. even) when ϕ is an odd (resp. even) permutation. To address whether the permutation groups generated by all complete mappings of a finite group G are “large” and primitive, Bors and Wang (2023) initiated the investigation into which groups admit both even and odd complete mappings. It is conjectured that such groups exist if and only if their Sylow 2-subgroups are trivial or noncyclic, except for some groups of small order. This paper reduces the problem of determining whether an arbitrary finite group admits both even and odd complete mappings to examining (1) the existence of complete mappings that stabilize a Sylow 2-subgroup for every finite simple group of Lie type and every sporadic simple group in the set <math><mo>{</mo><mrow><mi>Ly</mi></mrow><mo>,</mo><msub><mrow><mi>Co</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mspace></mspace><mi>B</mi><mo>,</mo><mspace></mspace><mrow><mi>Th</mi></mrow><mo>,</mo><mspace></mspace><msub><mrow><mi>Fi</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>Fi</mi></mrow><mrow><mn>23</mn></mrow></msub><mo>,</mo><mspace></mspace><msubsup><mrow><mi>Fi</mi></mrow><mrow><mn>24</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><mspace></mspace><mrow><mi>Ru</mi></mrow><mo>,</mo><msup><mrow><mi>O</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>N</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>Co</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>Co</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mrow><mi>HN</mi></mrow><mo>,</mo><mi>M</mi><mo>}</mo></math>, and (2) the existence of complete mappings that stabilize a subgroup of order <math><mn>8</mn><msubsup><mrow><mi>p</mi></mrow><mrow><mn>0</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msubsup></math> (for some odd prime <math><msub><mrow><mi>p</mi></mrow><mrow><mn>0</mn></mrow></msub></math> and <math><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>⩾</mo><mn>1</mn></math>) for every Ree group <math><mmultiscripts><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><none></none><mprescripts></mprescripts><none></none><mrow><mn>2</mn></mrow></mmultiscripts><mo>(</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></math> with <math><mi>n</mi><mo>⩾</mo><mn>1</mn></math>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"221 ","pages":"Article 106143"},"PeriodicalIF":1.2,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685864","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}

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 : 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

Towards odd-sunflowers: Temperate families and lightnings 走向奇异的向日葵：温带科和闪电

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2025-11-28 DOI: 10.1016/j.jcta.2025.106133

Jan Petr , Pavel Turek

Motivated by odd-sunflowers, introduced recently by Frankl, Pach, and Pálvölgyi, we initiate the study of temperate families: a family

F \subseteq P ([n])

is said to be temperate if each

A \in F

contains at most

| A |

elements of

F

as a proper subset.

We show that the maximum size of a temperate family is attained by the middle two layers of the hypercube

{0, 1}^{n}

. As a more general result, we obtain that the middle

t + 1

layers of the hypercube maximise the size of a family

F

such that each

A \in F

contains at most

\sum_{j = 1}^{t} (\begin{matrix} | A | \\ j \end{matrix})

elements of

F

as a proper subset. Moreover, we classify all such families consisting of the maximum number of sets.

In the case of intersecting temperate families, we find the maximum size and classify all intersecting temperate families consisting of the maximum number of sets for odd n. We also conjecture the maximum size for even n.

受Frankl、Pach和Pálvölgyi最近引入的奇向日葵的启发，我们发起了对温带家族的研究：如果每个a∈F最多包含F的|00个a |个元素作为适当子集，则称一个家族F∈P（[n]）是温带的。我们证明了温带族的最大尺寸是在超立方体{0,1}n的中间两层。作为一个更一般的结果，我们得到了超立方体的中间t+1层使族F的大小最大化，使得每个a∈F最多包含F的∑j=1t（| a |j）个元素作为一个固有子集。此外，我们对所有由最大数量集合组成的族进行了分类。对于相交的温带族，我们找到了最大的大小，并对奇数n下由最大集合数组成的所有相交的温带族进行了分类。我们还推测了偶数n下的最大大小。

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引用次数: 0

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 : 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

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