Damir Yeliussizov
It is a classical fundamental result that Schur-positive specializations of the ring of symmetric functions are characterized via totally positive functions whose parametrization describes the Edrei-Thoma theorem. In this paper we study positive specializations of symmetric Grothendieck polynomials, $K$-theoretic deformations of Schur polynomials.
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Buch and Rimányi proved a formula for specializations of double Grothendieck polynomials based on the Yang-Baxter equation related to the degenerate Hecke algebra. A geometric proof was found by Yong and Woo by constructing a Gröbner basis for the Kazhdan-Lusztig ideals. In this note, we give an elementary proof for this formula by using only divided difference operators.
Principal specializations of Grothendieck polynomials
本文研究了不包含1423模式的排列的Grothendieck多项式主专化，证明其可非负表示模式出现次数，并进一步推导出避免1342模式的非负展开，部分解决Gao、Meśzáros--Tanjaya和Dennin的猜想。
There has been recent interest in lower bounds for the principal specializations of Schubert polynomials $ν_w := \mathfrak S_w(1,\dots,1)$. We prove a conjecture of Yibo Gao in the setting of $1243$-avoiding permutations that gives a lower bound for $ν_w$ in terms of the permutation patterns contained in $w$. We extended this result to principal specializations of $β$-Grothendieck polynomials $ν^{(β)}_w := \mathfrak G^{(β)}_w(1,\dots,1)$ by restricting to the class of vexillary $1243$-avoiding permutations. Our methods are bijective, offering a combinatorial interpretation of the coefficients …
There is a remarkable formula for the principal specialization of a type A Schubert polynomial as a weighted sum over reduced words. Taking appropriate limits transforms this to an identity for the backstable Schubert polynomials recently introduced by Lam, Lee, and Shimozono. This note identifies some analogues of the latter formula for principal specializations of Schubert polynomials in classical types B, C, and D. We also describe some more general identities for Grothendieck polynomials. As a related application, we derive a simple proof of a pipe dream formula for involution Grothendieck…
Positive specializations of K-theoretic Schur P- and Q-functions
本文研究了K-理论Schur P-和Q函数的正专有化问题，并扩展了Edrei-Thoma定理的移位版本，同时探讨了极值调和函数的应用。
Decompositions of Grothendieck Polynomials
我们调查长期存在的问题，找到一个组合规则的舒伯特结构常数的$K$ -理论的旗品种（类型$A$）。本文给出了A.拉斯科-M P. Schützenberger（1982）作为$K$理论舒伯特类的多项式代表;然而，在格拉斯曼情形之外，没有关于它们乘法的正规则。给出了一个新的多项式基，给出了Grothendieck多项式在滑动基上展开的正组合公式，并给出了Grothendieck多项式在滑动基上展开乘积的正组合Littlewood-Richardson规则.我们的方法很容易推广到S的$β$ -Grothendieck多项式。福明A Kirillov（1994），在联结$K$ -理论中表示类，我们在这个更一般的背景下陈述我们的结果。滑动基的特殊化恢复了S的基本滑动多项式。Assaf-D Searles（2016），它在旗型品种的Chow环中扮演类似的角色。此外，滑动多项式的另一特殊化的稳定极限是T。Lam-P. Pylyavskyy（2007）多基本拟对称函数的基，$K$ -I. Gessel（1984）的基本拟对称函数。那些本身是拟对称的滑动多项式是多基本拟对称函数的截断，并形成拟对称多项式的基。
Inhomogeneous $q$-Whittaker polynomials II: ring theorem and positive specializations
本文研究非均匀$q$-Whittaker多项式，证明其构成特定交换环的基，扩展对称函数环，并探讨正特化与Macdonald正特化的联系，以及由此获得的概率分布。
Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and then derive various applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also give a new explanation of the finite expansion property for products of Grothendieck polynomials.
Newton polytopes and symmetric Grothendieck polynomials
对称Grothendieck多项式是在组合$K$理论中出现的Schur多项式的非齐次版本。一个多项式具有饱和牛顿多面体（SNP），如果在多面体中的每个格点是一个指数向量。我们证明了这些Grothendieck多项式的牛顿多面体及其齐次分量具有SNP。此外，每个齐次分支的牛顿多面体是置换面体。这解决了C. Monical-N。托坎-A Yong和A.芬克-K Mészáros-A.圣迪塞尔在这个特殊的情况下。
Grothendieck polynomials via permutation patterns and chains in the Bruhat order
本文给出了两类Grothendieck多项式的新公式。一种类型将Grothendieck多项式在至少两组变量中的任何特化表示为每组变量中Grothendieck多项式的乘积的线性组合，其中系数为Grothendieck多项式的舒伯特结构常数。另一种类型是根据布鲁哈特顺序的链。我们比较这第二种类型的其他结构的Grothendieck多项式在更一般的情况下，双Grothendieck多项式和密切相关的H-多项式。我们的方法是基于排列模式的几何。
The dual stable Grothendieck polynomials $g_λ$ and their sums $\sum_{μ\subsetλ} g_μ$ (which represent $K$-homology classes of boundary ideal sheaves and structure sheaves of Schubert varieties in the Grassmannians) have the same product structure constants. In this paper we first explain that the ring automorphism $g_λ\mapsto\sum_{μ\subsetλ} g_μ$ on the ring of symmetric functions is described as the operator $F^\perp$, the adjoint of the multiplication $(F\cdot)$, by a "group-like" element $F=\sum_{i} h_i$ where $h_i$ is the complete symmetric function. Next we give a generalization: starting…
Castelnuovo-Mumford regularity of skew-symmetric matrix Schubert varieties
本文通过给出辛Grothendieck多项式次数的组合公式，计算了斜对称矩阵Schubert簇的Castelnuovo-Mumford正则度，并刻画了其最高次齐次分量及最大正则度。
We prove the classification of homomorphisms from the algebra of symmetric functions to $\mathbb{R}$ with non-negative values on Macdonald symmetric functions $P_λ$, that was conjectured by S.V. Kerov in 1992.
$P$-Schur positive $P$-Grothendieck Polynomials
本文定义弱对称 $P$-Grothendieck 多项式以推广 $P$-Schur 多项式，并引入移位多重集表给出其组合实现，从而类比集值表推广半标准 Young 表的方式。
Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we prove tha…
Grothendieck bialgebras, Partition lattices and symmetric functions in noncommutative variables
证明了分划格代数的半塔的Grothendieck双代数与非交换变量对称函数双代数的分次对偶同构。特别是这种同构挑出一个典型的新的基础上的对称函数的非交换变量，这将是一个类似的舒尔函数的基础上，这双代数。
A note on the Grothendieck ring of the symmetric group
设$p$为素数，$n$为非零自然数。我们计算了代数$R\_n/pR\_n$的降Loewy级数，其中$R\_n$表示对称群$S\_n$的虚普通特征标环.
We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the K-theoretic analogue of that of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric fun…
Combinatorial interpretations of particular evaluations of complete and elementary symmetric functions
第一类和第二类Jacobi-Stirling数和Legendre-Stirling数是在[6]、[7]中首次提出的。本文指出Jacobi-Stirling数和Legendre-Stirling数是初等对称函数和完备对称函数的特殊化。然后，我们研究组合的解释，这种专业化，并获得新的组合解释的雅可比-斯特林和传奇-斯特林数。
Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability
该研究基于旗簇K理论中β=1的格罗滕迪克多项式，通过映射到TASEP描述了约化管道梦想随机置换的极限置换子及n^(1/3)阶Tracy-Widom GUE涨落，还解决了相关开放问题及β=1格罗滕迪克多项式主特殊化最大值问题。
Recently Galashin, Grinberg, and Liu introduced the refined dual stable Grothendieck polynomials, which are symmetric functions in $x=(x_1,x_2,\dots)$ with additional parameters $t=(t_1,t_2,\dots)$. The refined dual stable Grothendieck polynomials are defined as a generating function for reverse plane partitions of a given shape. They interpolate between Schur functions and dual stable Grothendieck polynomials introduced by Lam and Pylyavskyy in 2007. Flagged refined dual stable Grothendieck polynomials are a more refined version of refined dual stable Grothendieck polynomials, where lower and…
The symmetric Grothendieck polynomials representing Schubert classes in the $K$-theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type $A_n$ crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate o…
We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then provide a counterexample to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties and show that our work gives rise to an alternate explicit formula in these cases. We end with a new conjecture on the regularities of standard open patches of arbitrary Gr…
Lascoux expansion of the product of a Lascoux and a stable Grothendieck
本文利用Huang-Shimozono-Yu行插入算法，给出了Lascoux多项式与稳定Grothendieck多项式乘积展开为Lascoux多项式的组合表格公式，细化并推广了已有的K-理论Littlewood-Richardson规则及key-Schur展开。
The Schur polynomials $s_λ$ are essential in understanding the representation theory of the general linear group. They also describe the cohomology ring of the Grassmannians. For $ρ= (n, n-1, \dots, 1)$ a staircase shape and $μ\subseteq ρ$ a subpartition, the Stembridge equality states that $s_{ρ/μ} = s_{ρ/μ^T}$. This equality provides information about the symmetry of the cohomology ring. The stable Grothendieck polynomials $G_λ$, and the dual stable Grothendieck polynomials $g_λ$, developed by Buch, Lam, and Pylyavskyy, are variants of the Schur polynomials and describe the $K$-theory of the…
Kirillov's conjecture on Hecke-Grothendieck polynomials
本文利用统计力学中的代数方法，将多参数多项式表示为新一类可解格子模型的配分函数，证明了 Kirillov 关于 Hecke-Grothendieck 多项式正性猜想，并展示了更大类别的多项式中罕见的负系数实例。
The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous $K$-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model.
The main result of this paper is the introduction of marked graphs and the marked graph polynomials ($M$-polynomial) associated with them. These polynomials can be defined via a deletion-contraction operation. These polynomials are a generalization of the $W$-polynomial introduced by Noble and Welsh and a specialization of the $\mathbf{V}$-polynomial introduced by Ellis-Monaghan and Moffatt. In addition, we describe an important specialization of the $M$-polynomial which we call the $D$-polynomial. Furthermore, we give an efficient algorithm for computing the chromatic symmetric function of a …
Combinatorics of Double Grothendieck Polynomials
本文研究双 Grothendieck 多项式的组合结构，证明广义 Cauchy 恒等式，用三元表给出稳定情形的组合解释，构造半弱稳定插值并证明其在 $x=y$ 时的 $Q$-Schur 正性，最后提出关于阶乘 Schur $Q$-函数 $K$-理论类似物的猜想。
Grothendieck positivity for normal square root crystals
本文提出正规平方根晶体的组合框架，证明其特征是对称格罗滕迪克多项式的和，并给出布赫乘法规则的新证明。
Noncommutative Symmetric Functions Associated with a Code, Lazard Elimination, and Witt Vectors
Witt向量的泛环的构造与自由幺半群的Lazard因子分解通过一个非对易的类似物有关。这是通过将非交换对称函数的特殊化与代码相关联来完成的。
Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions
对偶稳定的格罗滕迪克多项式是舒尔函数的变形，起源于格拉斯曼的K-理论的研究。我们推广这些多项式通过引入一个可数家庭的额外的参数，我们证明了这种推广仍然定义对称函数。对于这一事实，我们给出了两个自包含的证明，其中一个构造了一个家庭的逆平面划分的集合上的对合推广的Bender-Knuth对合半标准tableaux，而另一个分类的结构的逆平面划分与项目1和2。
A Hopf algebra of parking functions
如果概率测度在$\R$上的矩被解释为完备齐次对称函数的特化，则其自由累积量直到符号是Schur正对称函数序列$(f_n)$的相应特化。证明了$(f_n)$是$\SG_n$在素停车函数集合上的自然置换表示的Frobenius特征.这种观察导致我们的停车功能，我们研究的一些细节的霍普夫代数的建设。
Narayana polynomials and Hall-Littlewood symmetric functions
我们表明，Narayana多项式是行Hall-Littlewood对称函数的特殊化。本文利用Koshy和Jonah关于Catalan数的公式，利用Koshy-环演算，将Koshy和Jonah关于Catalan数的公式推广到Narayana多项式。
Principal Specialization of Monomial Symmetric Polynomials and Group Determinants of Cyclic Groups
研究单项对称多项式在单位根处的主特殊化，给出显式公式，并证明这些特殊值自然出现在循环群行列式的k次幂展开中，推广了Ore公式，同时确定了循环群群积和k次幂的项数。
Diagrams of affine permutations, balanced labellings, and symmetric functions
本文从两个方面推广了Fomin，Greene，Reiner和Shimozono关于平衡标号的工作：（1）定义了仿射置换图和它们上的平衡标号;（2）定义了平衡标号的集值形式。我们证明了在仿射置换图上的列严格平衡标号产生Lam定义的仿射Stanley对称函数，并且列严格集值平衡标号产生Lam的仿射稳定Grothendieck多项式。此外，一旦我们施加适当的标记条件，有限置换图上的标记列严格集值平衡标号给出了Lascoux和Schützenberger的Grothendieck多项式的单项式展开。我们还给出了一个图是仿射置换图的充要条件。
Weak Bruhat interval modules of the 0-Hecke algebras for stable Grothendieck polynomials
本文针对稳定 Grothendieck 多项式的齐次分量，用标准集值杨表定义 0-Hecke 代数模，证明其拟对称特征等于该分量，并分解为弱 Bruhat 区间模的直和。
Noncommutative symmetric functions with matrix parameters
我们定义了新的家庭的非交换对称函数和准对称函数取决于两个矩阵的参数，更一般的参数与路径的二叉树。适当的专门化的两个矩阵，然后给回两个向量家庭的Hivert，Lascoux，和Thibon和非交换麦克唐纳功能的Bergeron和Zabrocki。
A contratableau model for K-theoretic Littlewood--Richardson rule
本文针对K-理论Littlewood-Richardson规则，给出基于集值可收缩表格的显式组合公式，推广了经典规则的可收缩表格模型，为计算对称Grothendieck多项式乘积系数提供新方法。
On flagged $K$-theoretic symmetric polynomials
本文利用松村的雅可比-特鲁迪型公式，给出了带标记斜格罗滕迪克多项式的费米子描述，并将其推广到作者先前关于斜格罗滕迪克多项式与多重舒尔函数的工作。
Skew stable Grothendieck polynomials are $K$-theoretic analogues of skew Schur polynomials. We give a free-fermionic presentation of skew stable Grothendieck polynomials and their dual symmetric functions. By using our presentation, we derive a family of determinantal formulas, which are $K$-analogues of the Jacobi-Trudi formula for skew Schur functions. We also introduce a combinatorial method to calculate certain expansions of skew (dual) stable Grothendieck polynomials by using the non-commutative supersymmetric Schur functions.
Differential Operator Specializations of Noncommutative Symmetric Functions
设$K$是任何有单位的交换$\mathbb Q$ -代数和$z=(z_1, ..., z_n)$交换或非交换自由变量.设$t$是一个形式参数，它与$z$和$K$的元素互换。我们用$\kzz$和$\kttzz$分别统一表示$z$在$K$和$K[[t]]$上的形式幂级数代数。对于任何$α\geq 1$，令$\cDazz$是由$\kzz$的微分算子生成的酉代数，这些微分算子至少使$z$的度增加$α-1$，而$ \ataz $是$\kttzz$与$o(H_t(z))\geq α$和$H_{t=0}(z)=0$的自同构群$F_t(z)=z-H_t(z)$。首先，对任何固定的$α\geq 1$和$F_t\in \ataz$，我们引入了$\kzz$的五个微分算子序列，并证明了它们的生成函数构成了微分代数$\cDazz$上的一个$\mathcal N$ CS（非交换对称）系统[Z4].因此，利用文献[GKLLRT]中首次引入的非交换对称函数（NCSF）的生成函数所构成的$\mathcal N$ CS系统的泛性质，我们得到了一族Hopf代数同态$\cS_{F_t}: {\mathcal N}Sym \to \cDazz$ $(F_t\in \ataz)$，当$F_t$满足一定条件时，这些同态也是保阶的.注意，上面的同态$\cS_{F_t}$也可以被看作是$\kzz$的微分算子对NCSF的特殊化。其次…
In this paper we study algebraic and combinatorial properties of Grothendieck polynomials and their dual polynomials by means of the Boson-Fermion correspondence. We show that these symmetric functions can be expressed as a vacuum expectation value of some operator that is written in terms of free-fermions. By using the free-fermionic expressions, we obtain alternative proofs of determinantal formulas and Pieri type formulas.
We introduce an enriched analogue of Lam and Pylyavskyy's theory of set-valued $P$-partitions. An an application, we construct a $K$-theoretic version of Stembridge's Hopf algebra of peak quasisymmetric functions. We show that the symmetric part of this algebra is generated by Ikeda and Naruse's shifted stable Grothendieck polynomials. We give the first proof that the natural skew analogues of these power series are also symmetric. A central tool in our constructions is a "$K$-theoretic" Hopf algebra of labeled posets, which may be of independent interest. Our results also lead to some new exp…
Superization and (q,t)-specialization in combinatorial Hopf algebras
我们扩展了一个经典的对称函数，超化过程，到几个组合霍普夫代数，并获得类似的钩内容公式的（q，t）-专业化的各种基地。利用树状结构产生特别是（q，t）-类似物的Bjorner-Wachs q-钩长度公式的二叉树，和类似的公式的平面树。
Spin invariant theory for the symmetric group
本文给出了自旋对称群在适当模中的不变量理论，其中涉及到多项式和外代数。我们解决了相应的分次多重性问题的Schur Q-函数和移位的q-钩公式的专业化。此外，我们还对Schur Q-函数的一个主特化公式给出了一个双射证明。
We express minors of Toeplitz matrices of finite and large dimension in terms of symmetric functions. Comparing the resulting expressions with the inverses of some Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skew Schur polynomials.
We give a $U_q(\mathfrak{sl}_n)$-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions, respectively. We show the result is isomorphic to a (generally infinite) direct sum of highest weight crystals, and for multiset-valued tableaux and valued-set tableaux, we provide an explicit bijection. As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions, which was not previously known. We also g…
In this paper, we consider a generalization of the Stirling number sequence of both kinds by using a specialization of a new family of symmetric functions. We give combinatorial interpretations for this symmetric functions by means of weighted lattice path and tilings. We also present some new convolutions involving the complete and elementary symmetric functions. Additionally, we introduce different families of set partitions to give combinatorial interpretations for the modular $s$-Stirling numbers.
A Murnaghan-Nakayama rule for Grothendieck polynomials of Grassmannian type
本文针对Grassmann流形K理论中的Grassmannian型Grothendieck多项式，建立了Murnaghan-Nakayama法则，用以将其与幂和对称多项式的乘积展开为Grothendieck多项式的线性组合。
A quasisymmetric function generalization of the chromatic symmetric function
图$G$的色对称函数$X_G$是由Stanley引入的.本文引入了一个拟对称的推广$X^k_G$，称为$G$的$k$ -色拟对称函数，并证明了它在拟对称函数的基本基上是正的.继专业化的$X_G$ $χ_G(λ)$，色多项式，我们还定义了一个推广$χ^k_G(λ)$，并表明，评价这个多项式的负值推广了一个定理的斯坦利有关非循环方向的色多项式。
Revisiting The Rédei-Berge Symmetric Functions via Matrix Algebra
本文通过矩阵代数方法重新研究有向图的Rédei-Berge对称函数，整合并扩展了其在幂和基与Schur基中的分解结果，并重新审视了哈密顿路径的相关结论。
The Equivalence of Two Graph Polynomials and a Symmetric Function
由于斯坦利的Tutte多项式的U-多项式、多色多项式和对称函数的推广，在它们中的任何一个的系数可以作为任何其他的系数的函数获得的意义上，已知是等价的。每个函数的定义都提出了一种自然的方法来概括它们，这也将Tutte的通用V函数作为一种特殊化。我们表明，等价性仍然是真的扩展功能，从而回答了多米尼克威尔士提出的一个问题。
Degrees of P-Grothendieck polynomials and regularity of Pfaffian varieties
本研究利用移位杨表组合数学证明P-Grothendieck多项式次数公式，结合已有结果得到vexillary斜对称矩阵Schubert簇的Castelnuovo-Mumford正则性上界，还得到Grassmannian相关多项式次数与簇正则性的新公式。
Tableau formula for vexillary double Edelman--Greene coefficients
本文针对 vexillary 置换情形，给出了双 β-Edelman–Greene 系数的表格式公式，使其 β-Graham 正性直接可见，并揭示了更精细的正性结构。
A Noncommutative Chromatic Symmetric Function
Stanley将一个图G与一个对称函数X_G联系起来，X_G在变量的某种特殊化下可约化为G的色多项式。然后，他证明了各种定理推广的结果约色多项式，以及新的，不能解释在这一水平。不幸的是，X_G不满足删除-压缩律，这使得它很难应用归纳法。我们在非交换变量中引入了一个对称函数，当变量允许交换时，它确实有这样一个规律，并且专门针对X_G。这使我们能够进一步推广一些斯坦利的定理，并证明他们在一个统一和直接的方式。此外，我们还对Stanley和Stembridge的（3+1）-free猜想作了一些改进.
Originally in 1954 the Tutte polynomial was a bivariate polynomial associated to a graph in order to enumerate the colorings of this graph and of its dual graph at the same time. However the Tutte polynomial reveals more of the internal structure of a graph, and contains even other specializations from other sciences like the Jones polynomial in Knot theory, the partition function of the Pott model in statistical physics, and the reliability polynomial in network theory. In this article, we study the Tutte polynomial associated to more general objects which are the arrangements of hyperplanes.…
The peak algebra and the Hecke-Clifford algebras at $q=0$
利用非交换对称函数的形式，我们导出了对称群的峰代数及其分次Hopf对偶的基本理论。我们的主要结果是在$q=0$上给出了Hecke-Clifford代数塔的峰代数及其分次对偶作为Grothendieck环的表示理论解释.
Classical double Grothendieck transitions
本文提出双Grothendieck多项式的递归公式，并应用该公式展开K-Stanley对称函数为正线性组合，同时解决若干正性猜想。
A q-analog of certain symmetric functions and one of its specializations
本文通过生成函数引入对称函数 $p_{n}^{(r)}$ 的 $q$-类似，并利用特殊化重新得到树逆序枚举多项式 $J_{n}^{(r)}$，进而导出其新递推、显式公式及森林统计量表示。
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