Advances in Applied Mathematics最新文献_第9页

q-Super Catalan numbers: Combinatorial identities, generating functions, and Narayana refinements q-超级加泰罗尼亚数：组合恒等式，生成函数，和Narayana细化

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2025-05-23 DOI: 10.1016/j.aam.2025.102911

Arthur Rodelet–Causse , Lenny Tevlin

We derive a number of combinatorial identities satisfied by the q-super Catalan numbers. In particular, we extend some of the known combinatorial identities (Touchard, Koshy, Reed Dawson) to the q-super Catalan numbers.

Next, we introduce some q-convolution identities involving q-central binomial and q-Catalan numbers, and derive a generating function for q-Catalan numbers.

Then we introduce Narayana-type refinements of the super Catalan numbers. We prove algebraically the γ-positivity of those refinements and give a combinatorial proof in a special case through the type B analog of noncrossing partitions. Then we introduce their natural q-analogs, prove their q-γ-positivity, and prove some identities they satisfy, generalizing identities of Kreweras [17] and Le Jen-Shoo [11]. Using yet another identity, we prove that these refinements are positive integer polynomials in q.

我们得到了一些由q-超加泰罗尼亚数满足的组合恒等式。特别地，我们将一些已知的组合恒等式（Touchard, Koshy, Reed Dawson）推广到q-超加泰罗尼亚数。其次，我们引入了一些涉及q-中心二项式和q-加泰罗尼亚数的q-卷积恒等式，并推导了q-加泰罗尼亚数的生成函数。然后我们引入了超加泰罗尼亚数的narayana型细化。我们从代数上证明了这些改进的γ-正性，并通过非交叉分区的B型模拟给出了一个特殊情况下的组合证明。然后引入它们的天然q-类似物，证明了它们的q-γ-正性，并证明了它们满足的一些恒等式，推广了Kreweras[17]和Le jen - sho[11]的恒等式。利用另一个恒等式，我们证明了这些改进是q的正整数多项式。

引用次数: 0

Equivalence classes of lower and upper descent weak Bruhat intervals 上下下降弱Bruhat区间的等价类

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2025-05-20 DOI: 10.1016/j.aam.2025.102910

Seung-Il Choi , Sun-Young Nam , Young-Tak Oh

<div><div>Let <math><mrow><mi>Int</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math> denote the set of nonempty left weak Bruhat intervals in the symmetric group <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math>. We investigate the equivalence relation <figure><img></figure> on <math><mrow><mi>Int</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math>, where <figure><img></figure> if and only if there exists a descent-preserving poset isomorphism between I and J. For each equivalence class C of <figure><img></figure>, a partial order ⪯ is defined by <math><msub><mrow><mo>[</mo><mi>σ</mi><mo>,</mo><mi>ρ</mi><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub><mo>⪯</mo><msub><mrow><mo>[</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math> if and only if <math><mi>σ</mi><msub><mrow><mo>⪯</mo></mrow><mrow><mi>R</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mo>′</mo></mrow></msup></math>. Kim–Lee–Oh (2024) showed that the poset <math><mo>(</mo><mi>C</mi><mo>,</mo><mo>⪯</mo><mo>)</mo></math> is isomorphic to a right weak Bruhat interval.</div><div>In this paper, we focus on lower and upper descent weak Bruhat intervals, specifically those of the form <math><msub><mrow><mo>[</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>,</mo><mi>σ</mi><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math> or <math><msub><mrow><mo>[</mo><mi>σ</mi><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math>, where <math><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo></math> is the longest element in the parabolic subgroup <math><msub><mrow><mi>S</mi></mrow><mrow><mi>S</mi></mrow></msub></math> of <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, generated by <math><mo>{</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mi>i</mi><mo>∈</mo><mi>S</mi><mo>}</mo></math> for a subset <math><mi>S</mi><mo>⊆</mo><mo>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo></math>, and <math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo></math> is the longest element among the minimal-length representatives of left <math><msub><mrow><mi>S</mi></mrow><mrow><mo>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo><mo>∖</mo><mi>S</mi></mrow></msub></math>-cosets in <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math>. We begin by providing a poset-theoretic characterization of the equivalence relation <figure><img></figure>. Using

设Int(n)表示对称群Sn中非空左弱Bruhat区间的集合。研究了Int(n)上的等价关系，其中当且仅当I与j之间存在一个保持下降的偏序同构。对于的每一个等价类C，一个偏阶⪯被定义为[σ，ρ]L⪯[σ ',ρ ‘]L当且仅当σ⪯Rσ ’。Kim-Lee-Oh（2024）证明了偏序集（C，⪯）与右弱Bruhat区间同构。本文主要研究下、上下降弱Bruhat区间，特别是[w0(S)，σ]L或[σ，w1(S)]L的下降弱Bruhat区间，其中w0(S)是Sn的抛物子群SS中的最长元素，由{si|i∈S}对一个子集S （n−1）产生，w1(S)是Sn中左S[n−1]∑S-集的最小长度代表中的最长元素。我们首先提供等价关系的位论表征。利用这一特征，当C是下下降区间或上下降区间时，可以确定等价类C中的最小和最大元素。在附加条件下，提供了（C，⪯）结构的详细描述。进一步，对于含有[w0(S)，σ]L的等价类，给出了B（[w0(S),σ]L）的一个内射壳，对于含有[σ，w1(S)]L的等价类，给出了B（[σ,w1(S)]L）的一个射影覆盖。其中，B(I)表示与I∈Int(n)相关联的0-Hecke代数的弱Bruhat区间模。应用所得结果研究了0-Hecke代数的射影不可分解模的商模和子模的下下降区间。

引用次数: 0

A result for hemi-bundled cross-intersecting families 半捆绑交叉族的结果

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2025-05-19 DOI: 10.1016/j.aam.2025.102912

Yongjiang Wu, Lihua Feng, Yongtao Li

Two families

F

and

G

are called cross-intersecting if for every

F \in F

and

G \in G

, the intersection

F \cap G

is non-empty. It is significant to determine the maximum sum of sizes of cross-intersecting families under the additional assumption that one of the two families is intersecting. Such a pair of families is said to be hemi-bundled. In particular, Frankl (2016) proved that for

k \geq 1, t \geq 0

and

n \geq 2 k + t

, if

F \subseteq (\begin{matrix} [n] \\ k + t \end{matrix})

and

G \subseteq (\begin{matrix} [n] \\ k \end{matrix})

are cross-intersecting families, in which

F

is non-empty and

(t + 1)

-intersecting, then

| F | + | G | \leq (\begin{matrix} n \\ k \end{matrix}) - (\begin{matrix} n - k - t \\ k \end{matrix}) + 1

. This bound is achieved when

F

consists of a single set. In this paper, we generalize this result under the constraint

| F | \geq r

for every

r \leq n - k - t + 1

. Moreover, we investigate the stability results of Katona's theorem for non-uniform families with the s-union property. Our result extends the stabilities established by Frankl (2017) and Li and Wu (2024). As applications, we revisit a recent result of Frankl and Wang (2024) as well as a result of Kupavskii (2018). Furthermore, we determine the extremal families in these two results.

如果对于每一个F∈F和G∈G，交集F∩G是非空的，那么两个族F和G被称为交叉交集。在两个族中有一个族相交的附加假设下，确定相交族的最大大小和是有意义的。这样的一对家庭被称为半捆绑。特别是Frankl（2016）证明了k≥1、t≥0、n≥2k+t时，如果F （[n]k+t）和G （[n]k）为交叉的家族，其中F为非空且（t+1）相交，则|F|+|G|≤(nk)−(n−k−tk)+1。当F由一个集合构成时，这个边界就实现了。本文在约束|F|≥r下，对每一个r≤n−k−t+1，推广了这一结果。此外，我们研究了具有s并性质的非一致族的卡托纳定理的稳定性结果。我们的结果扩展了Frankl（2017）和Li和Wu（2024）建立的稳定性。作为应用，我们回顾了Frankl和Wang（2024）的最新结果以及Kupavskii（2018）的结果。进一步，我们确定了这两个结果的极值族。

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引用次数: 0

Dissections of lacunary eta quotients and identically vanishing coefficients 空穴eta商和同消系数的剖分

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2025-04-30 DOI: 10.1016/j.aam.2025.102902

Tim Huber , James McLaughlin , Dongxi Ye