Lagrange Inversion and Schur Functions

Macdonald defined an involution on symmetric functions by considering the Lagrange inverse of the generating function of the complete homogeneous symmetric functions. The main result we prove in this note is that the images of skew Schur functions under this involution are either Schur positive or Schur negative symmetric functions. The proof relies on the combinatorics of Lagrange inversion. We also present a q-analogue of this result, which is related to the q-Lagrange inversion formula of Andrews, Garsia, and Gessel, as well as the operator of Bergeron and Garsia.     

Square -lattice paths and

The combinatorial -Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The -Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the 'th -Catalan number is the Hilbert series for the module of diagonal harmonic alternants in variables; it is also the coefficient of in the Schur expansion of . Using -analogues of labelled Dyck paths, Haglund et al. have proposed combinatorial conjectures for the monomial expansion of and the Hilbert series of the diagonal harmonics modules.

This article extends the combinatorial constructions of Haglund et al. to the case of lattice paths contained in squares. We define and study several -analogues of these lattice paths, proving combinatorial facts that closely parallel corresponding results for the -Catalan polynomials. We also conjecture an interpretation of our combinatorial polynomials in terms of the nabla operator. In particular, we conjecture combinatorial formulas for the monomial expansion of , the ``Hilbert series' , and the sign character .

     

Generalized harmonic numbers,Jacobi numbers and a Hankel determinant evaluation

In this paper we will introduce a sequence of complex numbers that are called the Jacobi numbers. This sequence generalizes in a natural way several sequences that are known in the literature, such as Catalan numbers, central binomial numbers, generalized catalan numbers, the coefficient of the Hilbert matrix and others. Subsequently, using a study of the polynomial of Jacobi, we give an evaluation of the Hankel determinants that associated with the sequence of Jacobi numbers. Finally, by finding a relationship between the Jacobi numbers and generalized harmonic numbers, we determine the evaluation of the Hankel determinants that are associated with generalized harmonic numbers.     

拉格朗日反演的q模拟与Bailey引理的注记

本文主要揭示了Gessel Ira.等给出的拉格朗日反演的q—模拟形式与An-drews G.E.的Bailey引理之间的相互转化的联系,做为例证,给出了利用这些关系得到的古典超几何级数(hypergeometric series)变换和求和公式的新证明,同时得到了模5、7、9、27四个新的Roger’s-Ramanujan类型的恒等式,其具有十分重要的组合意义。     

一类含有调和数的组合和

本文给出了含有调和数的一类组合和的求和公式．该公式有二熟知特例．     

与Bell多项式相关的一个新的矩阵反演及其应用

利用经典Lagrange反演公式, 本文给出了一个新的Bell矩阵反演, 由此建立了Bell多项式的一些新的性质, 其中包括一个Bell矩阵反演的封闭形式和经典Fa\`{a} di Bruno公式的一个逆形式.     

Inversion of Analytic Functions via Canonical Polynomials: A Matrix Approach

An alternative to Lagrange inversion for solving analytic systems is our technique of dual vector fields. We implement this approach using matrix multiplication that provides a fast algorithm for computing the coefficients of the inverse function. Examples include calculating the critical points of the sinc function. Maple procedures are included which can be directly translated for doing numerical computations in Java or C. A preliminary version of this paper has been presented at AISC 2006.     

一类精细化Eulerian多项式的若干性质

最近,孙华定义了一类新的精细化Eulerian多项式,即$$A_n(p,q)=\sum_{\pi\in \mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)},\ \ n\ge 1,$$ 其中$S_n$表示$\{1,2,\ldots,n\}$上全体$n$阶排列的集合, odes$(\pi)$与edes$(\pi)$分别表示$S_n$中排列$\pi$的奇数位与偶数位上降位数的个数.本文利用经典的Eulerian多项式$A_n(q)$ 与Catalan 序列的生成函数$C(q)$,得到精细化Eulerian 多项式$A_n(p,q)$的指数型生成函数及$A_n(p,q)$的显示表达式.在一些特殊情形,本文建立了$A_n(p,q)$与$A_n(0,q)$或$A_n(p,0)$之间的联系,并利用Eulerian数表示多项式$A_n(0,q)$的系数.特别地,这些联系揭示了Euler数$E_n$与Eulerian数$A_{n,k}$之间的一种新的关系.     

Lagrange Inversion: When and How

The aim of the present paper is to show how the Lagrange Inversion Formula (LIF) can be applied in a straight-forward way i) to find the generating function of many combinatorial sequences, ii) to extract the coefficients of a formal power series, iii) to compute combinatorial sums, and iv) to perform the inversion of combinatorial identities. Particular forms of the LIF are studied, in order to simplify the computation steps. Some examples are taken from the literature, but their proof is different from the usual, and others are new.      

Classroom note: A comparison of the method of least squares and the method of averages

Two techniques for determining a straight line fit to data are compared.     

一个全局和局部的Metro Carlo电磁转换及计算

本文提出了一个全局和局部的Metro Carlo电磁转换方法,找到了电磁材料参数,提供了一种关于全局和局部的电磁转换计算方法。     

Multivariate Interpolation Using Carlitz's Inversion Formulas

Multivariate rational exponential Lagrange interpolation formulas, Hermite interpolation formulas, and Hermite–Fejér interpolation formulas of the Newton type are established by using Carlitz's inversion formulas. The recurrence relation for constructing Lagrange interpolation is also given. In addition, by setting q1 in the obtained formulas, we obtain the corresponding polynomial interpolation formulas with combinatorial form.     

Lattice paths and the q-ballot polynomials

Two statistics with respect to “upper-corners” and “lower-corners” are introduced for lattice paths. The corresponding refined generating functions are shown to be closely related to the q-ballot polynomials that extend the well-known Narayana polynomials and Catalan numbers.     

Itoh-Tsujii Inversion in Standard Basis and Its Application in Cryptography and Codes

This contribution is concerned with a generalization of Itoh and Tsujii's algorithm for inversion in extension fields . Unlike the original algorithm, the method introduced here uses a standard (or polynomial) basis representation. The inversion method is generalized for standard basis representation and relevant complexity expressions are established, consisting of the number of extension field multiplications and exponentiations. As the main contribution, for three important classes of fields we show that the Frobenius map can be explored to perform the exponentiations required for the inversion algorithm efficiently. As an important consequence, Itoh and Tsujii's inversion method shows almost the same practical complexity for standard basis as for normal basis representation for the field classes considered.     

一类广义Catalan数及其应用

引进了一类广义Catalan数,并赋予这类广义Catalan数组合意义,用这类广义Catalan数得到一类不定方程的解数.     

Several formulas for special values of the Bell polynomials of the second kind and applications

In the paper, the authors establish several explicit formulas for special values of the Bell polynomials of the second kind, connect these formulas with the Bessel polynomials, and apply these formulas to give new expressions for the Catalan numbers and to compute arbitrary higher order derivatives of elementary functions such as the since, cosine, exponential, logarithm, arcsine, and arccosine of the square root for the variable.     

Some identities on the Catalan, Motzkin and Schröder numbers

In this paper, some identities between the Catalan, Motzkin and Schröder numbers are obtained by using the Riordan group. We also present two combinatorial proofs for an identity related to the Catalan numbers with the Motzkin numbers and an identity related to the Schröder numbers with the Motzkin numbers, respectively.     

Ballot matrix as Catalan matrix power and related identities

We use an analytical approach to find the kth power of the Catalan matrix. Precisely, it is proven that the power of the Catalan matrix is a lower triangular Toeplitz matrix which contains the well-known ballot numbers. A result from [H. S. Wilf, Generatingfunctionology, Academic Press, New York, 1990, Free download available from http://www.math.upenn.edu/~wilf/Downld.html.], related to the generating function for Catalan numbers, is extended to the negative integers. Three interesting representations for Catalan numbers by means of the binomial coefficients and the hypergeometric functions are obtained using relations between Catalan matrix powers.     

A Modification of Lagrange Interpolation

A modification of Lagrange interpolation based on the zeros of the Chebyshev polynomial of the second kind is constructed, which interpolates at many ofgiven data. Thus, for this node-system the main result gives an affimative answer to a problem suggested by Bernstein in 1930. Moreover, our modification has a Timan-Gopengauz type approximation rate.     

Bessel变换的一个下界

Bessel逆问题在物理、化学和工程学等诸多领域有重要应用.解决线性逆问题的传统方法不适合处理具有奇异性曲线边缘的二元函数.鉴于切波对这一类函数的最优表示能力,相关文献采用切波方法研究Bessel逆问题,构造了目标函数的切波域值估计器,得到了它在函数空间V中积分均方差收敛阶的上界.在此基础上利用统计理论给出其最小最大风险的一个下界,证明了在估计Bessel逆问题时此估计器是最优的.