Catalan numbers and Schubert polynomials for $w=1(n+1)... 2$
Alexander Woo
证明了当$w=1(n+1)...2$满足时，Schubert多项式S_w特殊化为Catalan数C_n.这个结果的几个证明以及一个q-模拟。给出了Schubert簇的奇异性的一个应用。
We show that the Schubert polynomial S_w specializes to the Catalan number C_n when $w=1(n+1)...2$. Several proofs of this result as well as a q-analog are given. An application to the singularities of Schubert varieties is given.
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Multiplication of a Schubert polynomial by a Stanley symmetric polynomial
我们证明，组合，由斯坦利对称多项式的舒伯特多项式的产品是一个截断的舒伯特多项式。利用Monk规则，我们得到了截断Schubert多项式的Schubert多项式展开式的一个非负组合公式。结合这些结果，我们给出了Schubert基上Schubert多项式与Schur多项式乘积的一个非负组合规则。
Quantum double Schubert polynomials represent Schubert classes
量子双舒伯特多项式研究基里洛夫和Maeno，并由Ciocan Fontanine和富尔顿，显示代表舒伯特类在金的介绍等变量子上同调的旗品种。我们定义了量子双舒伯特多项式的抛物型类似物，并证明了它们表示部分旗簇的等变量子上同调的舒伯特类。对于完全标志，Anderson和Chen用不同的方法给出了一个证明.
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Skew Schubert polynomials
我们定义斜舒伯特多项式的规范形式（多项式）的代表某些类的上同调旗流形。我们表明，这个定义扩展了最近的舒伯特多项式建设由于Bergeron和Sottile在某些增加标记链的Bruhat顺序的对称群。这些斜舒伯特多项式扩展的基础上的舒伯特多项式的非负整数系数，正是结构常数的上同调的复旗品种就其基础的舒伯特类。我们重新推导的Bergeron和Sottile在一个纯粹的组合方式的建设，有关它的建设舒伯特多项式的rc-图。
Kohnert proposed the first monomial positive formula for Schubert polynomials as the generating polynomial for certain unit cell diagrams obtained from the Rothe diagram of a permutation. Billey, Jockusch and Stanley gave the first proven formula for Schubert polynomials as the generating polynomial for compatible sequences of reduced words of a permutation. In this paper, we give an explicit bijection between these two models, thereby definitively proving Kohnert's rule for Schubert polynomials.
We survey the recent study of involution Schubert polynomials and a modest generalization that we call degenerate involution Schubert polynomials. We cite several conditions when (degenerate) involution Schubert polynomials have simple factorization formulae. Such polynomials can be computed by traversing through chains in certain weak order posets, and we provide explicit descriptions of such chains in weak order for involutions and degenerate involutions. As an application, we give several examples of how certain multiplicity-free sums of Schubert polynomials factor completely into very simp…
Schubert polynomials and Arakelov theory of orthogonal flag varieties
本文提出了一种组合显式Schubert多项式理论，它表示正交旗簇的上同调环的Borel表示中的Schubert类.我们使用这些多项式来描述这些空间上的算术舒伯特演算。给出了自然算术相交数的计算方法，并证明了它们都是有理数。
Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams
我们引入了两个新的多项式的基地，解除单项式和基本拟对称函数的完整多项式环。通过定义一个新的条件上的梦想，所谓的准山之内，我们给出了一个积极的组合规则扩展舒伯特多项式到这些新的基地，平行的扩张Schur函数到基本的准对称函数。结果，我们得到了Schubert多项式到Stanley对称函数的稳定极限的一个改进。我们还给出了这些基的正结构常数的组合规则，分别推广了拟混洗积和混洗积。我们用它来给一个Littlewood-理查森规则的舒伯特多项式的产品扩展到基本滑动多项式，并给出公式的产品的斯坦利对称函数的舒伯特结构常数。
Schubert polynomials and Arakelov theory of symplectic flag varieties
设X是辛群的旗簇。本文提出了一个组合显式Schubert多项式理论，它表示X的上同调环的Borel表示中的Schubert类。我们用这些多项式来描述X上的算术Schubert演算。此外，我们还给出了计算X上自然算术Chern数的方法，并证明了它们都是有理数。
Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other known bases including key polynomials and fundamental slide polynomials. We unify these and existing models by giving simple bijections between the combinatorial objects indexing each. In particular, we give a simple bijective proof that the balanced tableaux of Edelman and Greene enumerate reduced expressions and a direct combinatorial proof of Kohnert's algori…
From quantum Schubert polynomials to k-Schur functions via the Toda lattice
我们证明了Lapointe-Lascoux-Morse k-Schur函数（t=1）和Fomin-Gelfand-Postnikov量子Schubert多项式可以通过一个有理代换相互得到.这是基于Kostant的户田格的解决方案和彼得森的工作量子舒伯特演算。
Orthogonal divided differences and Schubert polynomials, P-functions, and vertex operators
本文给出了几个新的公式，这些公式对于与正交群及相关的正交退化轨迹有关的Schubert演算是有用的。
From double quantum Schubert polynomials to k-double Schur functions via the Toda lattice
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Equivariant quantum Schubert polynomials
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The Prism tableau model for Schubert polynomials
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Let $Φ$ be a root system. Postnikov introduced and studied the mixed $Φ$-Eulerian numbers. These numbers indicate the mixed volumes of $Φ$-hypersimplices. As specializations of these numbers, one can obtain the usual Eulerian numbers, the Catalan numbers, and the binomial coefficients. Recent work of Berget-Spink-Tseng gave a simple computation for the mixed $Φ$-Eulerian numbers when $Φ$ is of type $A$. In this paper we connect a relation between mixed $Φ$-Eulerian numbers and Peterson Schubert calculus. By using the connection, we provide a combinatorial model for the computation of Berget-Sp…
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Quantum Bruhat graph and Schubert polynomials
量子Bruhat图是由Bruhat序覆盖关系构成的图的推广，它与G/B的量子上同调环有着天然的联系。我们增强了富尔顿和伍德沃德的结果表明，最小单项式的量子参数，发生在量子产品的两个舒伯特类有一个简单的解释，在这个图中的有向路径。我们定义路径舒伯特多项式，这是量子上同调的类似物最近推出的Lenart和Sottile的斜舒伯特多项式。它们由A型量子Bruhat图中的路和给出。旗流形的3点Gromov-Witten不变量用这些多项式表示。这种构造给出了两个舒伯特类的量子乘积中出现的量子参数中的所有单项式的集合的组合描述。
Double Schubert polynomials for the classical groups
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Schubert polynomials for the affine Grassmannian
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Ikeda-Mihalcea-Naruse's double Schubert polynomials represent the equivariant cohomology classes of Schubert varieties in the type C flag varieties. The goal of this paper is to obtain a new tableau formula of these polynomials associated to vexillary signed permutations introduced by Anderson-Fulton. To achieve that goal, we introduce flagged factorial (Schur) $Q$-functions, combinatorially defined in terms of (flagged) marked shifted tableaux for flagged strict partitions, and prove their Schur-Pfaffian formula. As an application, we also obtain a new combinatorial formula of factorial $Q$-f…
Schubert Polynomials and Quiver Formulas
Buch和Fulton的工作建立了一个与$A$型定向轨道相关的一般类型简并轨迹的公式。这个公式中的主要成分是Schur行列式和某些整数，系数，它推广了经典的Littlewood-Richardson系数。本文的目的是证明当定义退化轨迹的秩条件由置换给出时，系统系数的一个正的组合公式。特别是，这给新的扩展富尔顿的普遍舒伯特多项式和舒伯特多项式的Lascoux和Schützenberger。
Schubert Polynomials for the affine Grassmannian of the symplectic group
研究了辛群的仿射格拉斯曼群的Schubert演算。Gr的整同调环和上同调环被定义为对称函数的对偶霍普夫代数，并由Schur的P-函数和Q-函数定义.一个明确的组合描述Gr的上同调的Schubert基，这是扩展到仿射型C斯坦利对称函数的定义。对于一个特殊的Schubert类与一个任意Schubert类的乘积，给出了一个同调Pieri规则。
Schubert polynomials and degeneracy locus formulas
在我们以前的工作arXiv：1305.3543中，我们采用了Fomin，Stanley和Kirillov对舒伯特多项式的方法来获得简单，统一的证明，即Lascoux和Schutzenberger以及Ikeda，Mihalcea和Naruse的双舒伯特多项式表示Fulton意义下的典型群的退化轨迹。以此为出发点，用纯组合的方法，我们得到了arXiv：0908.3628的通式的一个新的证明，它表示来自任何各向同性部分旗簇的退化轨迹。在此过程中，我们还发现了几个新的公式，并阐明了一些早期公式之间的联系。
Schubert Calculus, Schubert Cell, Schubert Cycle, and Schubert Polynomials
我们简要介绍了四个主题：舒伯特微积分，舒伯特细胞，舒伯特循环和舒伯特多项式。
In this paper, we give a formula for the number of lattice points in the dilations of Schubert matroid polytopes. As applications, we obtain the Ehrhart polynomials of uniform and minimal matroids as special cases, and give a recursive formula for the Ehrhart polynomials of $(a,b)$-Catalan matroids. Ferroni showed that uniform and minimal matroids are Ehrhart positive. We show that all sparse paving Schubert matroids are Ehrhart positive and their Ehrhart polynomials are coefficient-wisely bounded by those of minimal and uniform matroids. This confirms a conjecture of Ferroni for the case of s…
Quantum and affine Schubert calculus and Macdonald polynomials
我们明确地建立了对称Macdonald多项式的理论与量子和仿射Schubert演算对齐使用的发现，区别弱链可以识别的链在强（Bruhat）阶偏序集上的类型- $A$仿射外尔群。我们构造了两个单参数函数族，它们分别与Hall-Littlewood和Macdonald的$P$ -函数正迁移，并专门研究了仿射格拉斯曼的Schubert类同调和上同调的代表.我们的方法使我们猜想，在一个定义的3点属0 Gromov-Witten不变量旗流形的所有元素可以制定为强覆盖。
A formula for K-theory truncation Schubert calculus
将$\{x_1,x_2,x_3,...\}$中多项式$p$的“截断”$r_{t}(p)$定义为除了第一个$t$变量之外的所有变量都设置为零的多项式。在某些好的情况下，舒伯特或格罗滕迪克多项式的截断可以再次是舒伯特或格罗滕迪克多项式。我们利用这一现象给出了$K(Flags({\mathbb C}^n))$中某些Schubert结构常数的免减公式，特别是推广了[Kogan '00]中仅处理上同调的公式和[Buch' 02]中关于Grassmannian情况的公式。答案的项是使用置换图上的“marching”运算来计算的。
Infinite flags and Schubert polynomials
本文借助无限维旗簇与退化轨迹几何研究 Schubert 多项式，证明相关等变余积系数的 Graham 正性并给出多项式展开，同时构造 C 型旗簇嵌入及其上同调拉回映射。
Double Schubert polynomials and degeneracy loci for the classical groups
本文提出了B，C，D型李的双Schubert多项式P_w（X，Y）理论，它自然地推广了A型Schutzenberger的Lascoux多项式族。这些多项式满足正性、正交性和稳定性，并代表了向量丛的Schubert簇和退化轨迹类。当w是外尔群的极大Grassmannian元时，P_w（X，Y）可以用Schur型行列式和Pfirmans表示，类似于Kempf和Laksov的A型公式.由量子上同调激发的一个例子表明，丛的“各向同性态射”的退化轨迹不存在陈类公式。
A Unified Approach to Combinatorial Formulas for Schubert Polynomials
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Schubert Polynomials and $k$-Schur functions
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We examine the relationship between the (double) Schubert polynomials of Billey-Haiman and Ikeda-Mihalcea-Naruse and the (double) theta and eta polynomials of Buch-Kresch-Tamvakis and Wilson from the perspective of Weyl group invariants. We obtain generators for the kernel of the natural map from the corresponding ring of Schubert polynomials to the (equivariant) cohomology ring of symplectic and orthogonal flag manifolds.
In this paper we study a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials $f_{i,j}$ were introduced by Abe-Harada-Horiguchi-Masuda. We show that every polynomial $f_{i,j}$ is an alternating sum of certain Schubert polynomials.
We give bijective proofs of Monk's rule for Schubert and double Schubert polynomials computed with bumpless pipe dreams. In particular, they specialize to bijective proofs of transition and cotransition formulas of Schubert and double Schubert polynomials, which can be used to establish bijections with ordinary pipe dreams.
Polynomiality of Stretched Schubert Structure Constants and Key Coefficients
研究关键多项式和舒伯特多项式仿射族中单项式系数的多项式性，通过结合德马祖尔算子等方法证明其最终为多项式，并扩展到有限乘积，利用舒伯特对偶性证明拉伸舒伯特结构常数最终是多项式，解决相关猜想。
Pieri's rule for flag manifolds and Schubert polynomials
在旗流形的整上同调环中，我们建立了与一个特殊Schubert簇的类相乘的公式。这个公式还描述了舒伯特多项式与初等对称多项式或完全齐次对称多项式的乘法。因此，我们推广了经典的Pieri规则对称多项式/格拉斯曼品种舒伯特多项式/旗流形。我们的主要技术是一个明确的几何描述的某些交叉的舒伯特品种。这种方法使我们能够计算额外的结构常数的上同调环，我们表示在对称群上的Bruhat顺序的路径。
Schubert polynomials and classes of Hessenberg varieties
正则半单Hessenberg簇是旗簇的一族子簇，出现在数论、数值分析、表示论、代数几何和组合学中。给出了正则半单Hessenberg簇类用Chern类表示的Giambelli公式。事实上，我们表明，上同调类的每一个经常半单Hessenberg品种是特定的双舒伯特多项式的特殊化，给这样的特殊化一个自然的几何解释。我们还分解这样的类的Schubert基础的上同调环的标志品种。得到的系数是非负的，我们给出了封闭的组合公式的系数在许多情况下。我们引入了一个密切相关的家庭计划称为定期幂零Hessenberg计划，并使用我们的结果来确定当这些计划减少。
A Rational Catalan Formula for $(m,3)$-Hikita Polynomials
基于最近的$(3,m)$ -加泰罗尼亚多项式的公式，我们描述了一个公式$(3,m)$ -Hikita多项式的加泰罗尼亚多项式。该公式揭示了Hikita多项式系数间的一种令人惊讶的关系，蕴含了更深层次的递归关系，并证明了$(3,m)$ -Hikita多项式的$q,t$ -对称性.
Exercises for A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode
本文为“多项式方程的超加泰罗尼亚级数解与测地线”一文配套编写习题并给出解答，以帮助读者理解和练习相关内容。
Graham positivity of triple Schubert calculus
本文通过建立精细化 Graham 正性定理，证明 Samuel 关于三组变量中两个双 Schubert 多项式展开系数正性的猜想，并推出 Kirillov 关于斜除以差分算子作用于 Schubert 多项式正性的猜想。
Tuenter polynomials and a Catalan triangle
我们认为Tuenter多项式的下降因子的线性组合，并表明，这些线性组合的系数表示通过一个加泰罗尼亚三角形的数字。我们还描述了一个三角形的系数在一些多项式。
A Molev-Sagan type formula for double Schubert polynomials
本文提出计算双舒伯特多项式乘积的 Molev-Sagan 型公式及 Pieri 公式，证明其正性并给出格拉斯曼流形的新等变 Littlewood-Richardson 规则，同时组合证明了 Kirillov 关于斜舒伯特多项式系数非负性的猜想。
q-Catalan numbers and q-Narayana polynomials
本文证明了Catalan数的各种自然q-模拟可以用统一的方法得到。此外，我们还计算了它们的汉克尔行列式。
A Demazure crystal construction for Schubert polynomials
Stanley对称函数是Schubert多项式的稳定极限。在本文中，我们表明，相反，舒伯特多项式是Demazure截断斯坦利对称函数。这类似于一般线性群的Schur函数和Demazure特征标之间的关系。我们建立了这种连接，通过施加Demazure晶体结构的关键tableaux，最近介绍了第一作者在连接Demazure字符和舒伯特多项式，并将其链接到A型晶体结构减少字分解，最近介绍了莫尔斯和第二作者在连接斯坦利对称函数。
In this note, we study two generalizations of the Catalan numbers, namely the $s$-Catalan numbers and the spin $s$-Catalan numbers. These numbers first appeared in relation to quantum physics problems about spin multiplicities. We give a combinatorial description for these numbers in terms of Littlewood-Richardson coefficients, and explain some of the properties they exhibit in terms of Littlewood-Richardson polynomials.
We prove that the inverse of the Hankel matrix of the reciprocals of the Catalan numbers has integer entries. We generalize the result to an infinite family of generalized Catalan numbers. The Hankel matrices that we consider are associated with orthogonal polynomials that are variants of Jacobi polynomials. Our proofs use these polynomials and computer algebra based on Wilf-Zeilberger theory.
We prove that an inclusion-exclusion inspired expression of Schubert polynomials of permutations that avoid the patterns 1432 and 1423 is nonnegative. Our theorem implies a partial affirmative answer to a recent conjecture of Yibo Gao about principal specializations of Schubert polynomials. We propose a general framework for finding inclusion-exclusion inspired expression of Schubert polynomials of all permutations.
The closure of a generic torus orbit in the flag variety $G/B$ of type $A$ is known to be a permutohedral variety and its Poincare polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincare polynomial of a generic torus orbit closure in a Schubert variety in $G/B$. When the generic torus orbit closure in a Schubert variety is smooth, its Poincare polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.
Vanishing of Schubert coefficients in probabilistic polynomial time
本文从算法角度完整解决了舒伯特消失问题，证明舒伯特结构常数是否为零可在概率多项式时间内判定。
We use the inversion of coefficient arrays to define dual polynomials to the Fibonacci and Catalan-Fibonacci polynomials, and we explore the properties of these new polynomials sequences. Many of the arrays involved are Riordan arrays. Direct links to the counting of Motzkin paths by different statistics emerge.
On super Catalan polynomials
我们提出了一个超Catalan数$(2m)!(2n)!/2m!n!(m+n)!$的$q$ -模拟，它也推广了由于Fürlinger和Hofbauer，对于$λ=0$和$λ=1$的$q$ -Catalan数$c_n(λ)$。当$m=2$。
Schubert polynomials, 132-patterns, and Stanley's conjecture
受R. P. Stanley给出了舒伯特多项式的系数之和在$132$ -模式包容下的下界。
The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by Knutson and Miller (2005) via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics for the class of "vexillary'' matrix Schubert varieties. We initiate a study of general diagonal degenerations, relating them to a neglected formula of Lascoux (2002) in terms of the $6$-vertex ice model (recently rediscovered by Lam, Lee, and S…
Connection between Schubert polynomials and top Lascoux polynomials
本文通过一个简单算子建立Schubert多项式与top Lascoux多项式的联系，证明二者结构常数相同，并转化若干组合公式结果。
Identities involving Narayana polynomials and Catalan numbers
我们首先建立了Narayana多项式可以表示为勒让德多项式的积分的结果。然后我们用三个不同的恒等式表示卡塔兰数。我们给出了三个不同的证明，即两个代数证明和一个组合证明。一些应用程序也导致许多已知的和新的身份。
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