Restricted 132-Alternating Permutations and Chebyshev Polynomials

A permutation is said to be alternating if it starts with rise and then descents and rises come in turn. In this paper we study the generating function for the number of alternating permutations on n letters that avoid or contain exactly once 132 and also avoid or contain exactly once an arbitrary pattern on k letters. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.AMS Subject Classification: 05A05, 05A15, 30B70, 42C05.     

A New Class of Wilf-Equivalent Permutations

For about 10 years, the classification up to Wilf equivalence of permutation patterns was thought completed up to length 6. In this paper, we establish a new class of Wilf-equivalent permutation patterns, namely, (n – 1, n – 2, n, ) (n – 2, n, n – 1, ) for any S _{n –3}. In particular, at level n = 6, this result includes the only missing equivalence (546213) (465213), and for n = 7 it completes the classification of permutation patterns by settling all remaining cases in S ₇.     

On the restricted Chebyshev–Boubaker polynomials

Using the language of Riordan arrays, we study a one-parameter family of orthogonal polynomials that we call the restricted Chebyshev–Boubaker polynomials. We characterize these polynomials in terms of the three term recurrences that they satisfy, and we study certain central sequences defined by their coefficient arrays. We give an integral representation for their moments, and we show that the Hankel transforms of these moments have a simple form. We show that the (sequence) Hankel transform of the row sums of the corresponding moment matrix is defined by a family of polynomials closely related to the Chebyshev polynomials of the second kind, and that these row sums are in fact the moments of another family of orthogonal polynomials.     

Permutations Containing Many Patterns

It is shown that the maximum number of patterns that can occur in a permutation of length n is asymptotically 2 ⁿ . This significantly improves a previous result of Coleman. Received September 28, 2006     

避免$2$字母符号模式的具有偶数个符号的带符号排列

Let Dn be the set of all signed permutations on [n] = {1,... ,n} with even signs, and let ：Dn（T） be the set of all signed permutations in Dn which avoids a set T of signed patterns. In this paper, we find all the cardinalities of the sets Dn（T） where T B2. Some of the cardinalities encountered involve inverse binomial coefficients, binomial coefficients, Catalan numbers, and Fibonacci numbers.     

Enumerating Permutations Avoiding Three Babson-Steingrímsson Patterns

We settle some conjectures formulated by A. Claesson and T. Mansour concerning generalized pattern avoidance of permutations. In particular, we solve the problem of the enumeration of permutations avoiding three generalized patterns of type (1, 2) or (2, 1) by using ECO method and a graphical representation of permutations.Received July 15, 2004     

Certain bilateral generating relations for generalized hypergeometric functions

Recently, we introduced a class of generalized hypergeometric functionsI n:(b _q)/α:(a _p) (x, w) by using a difference operator Δ _x,w , where . In this paper an attempt has been made to obtain some bilateral generating relations associated withI _n ^ga (x, w). Each result is followed by its applications to the classical orthogonal polynomials.     

第二类Chebyshev多项式系数的绝对值和的卷积

本文主要的目的是利用广义Fibonacci多项式的生成函数及其偏导数来研究第二类Cheby-shev多项式卷积的计算,并给出—个有趣的计算公式.     

Enumeration of Decomposable Combinatorial Structures with Restricted Patterns

Decomposable combinatorial structures are studied with restricted patterns. We focus on the decomposable structures in the exp-log class. Using the method of analysis of singularities introduced by Flajolet and Odlyzko [5], we provide an estimate for the probability that a decomposable structure of size n has a given restricted pattern. We exemplify with several decomposable structures like permutations and polynomials over finite fields.      

A Tau Method with Perturbed Boundary Conditions for Certain Ordinary Differential Equations

Ortiz recursive formulation of the Lanczos Tau method (TM) is a powerful and efficient technique for producing polynomial approximations for initial or boundary value problems. The method consists in obtaining a polynomial which satisfies (i) a perturbed version of the given differential equation, and (ii) the imposed supplementary conditions exactly. This paper introduces a new form of the TM, (denoted by PTM), for a restricted class of differential equations, in which the differential equations as well as the supplementary conditions are perturbed simultaneously. PTM is compared to the classical TM from the point of view of their errors: it is found that the PTM error is smaller and more oscillatory than that of the TM; we further find that approximations nearly as accurate as minimax polynomial approximations can be constructed by means of the PTM. Detailed formulae are derived for the polynomial approximations in TM and PTM, based on Canonical Polynomials. Moreover, various limiting properties of Tau coefficients are established and it is shown that the perturbation in PTM behaves asymptotically proportional to a Chebyshev polynomial.     

Generating functions and companion symmetric linear functionals

In this contribution we analyze the generating functions for polynomials orthogonal with respect to a symmetric linear functional u, i.e., a linear application in the linear space of polynomials with complex coefficients such that . In some cases we can deduce explicitly the expression for the generating function

Some Combinatorics Related to Central Binomial Coefficients: Grand-Dyck Paths, Coloured Noncrossing Partitions and Signed Pattern Avoiding Permutations

We give some interpretations to certain integer sequences in terms of parameters on Grand-Dyck paths and coloured noncrossing partitions, and we find some new bijections relating Grand-Dyck paths and signed pattern avoiding permutations. Next we transfer a natural distributive lattice structure on Grand-Dyck paths to coloured noncrossing partitions and signed pattern avoiding permutations, thus showing, in particular, that it is isomorphic to the structure induced by the (strong) Bruhat order on a certain set of signed pattern avoiding permutations.     

Gaussian integration of Chebyshev polynomials and analytic functions

Explicit bounds for the quadrature error of thenth Gauss-Legendre quadrature rule applied to themth Chebyshev polynomial are derived. They are precise up to the orderO(m ⁴ n ^–6). As an application, error constants for classes of functions which are analytic in the interior of an ellipse are estimated. The location of the maxima of the corresponding kernel function is investigated.Dedicated to Luigi Gatteschi on the occasion of his 70th birthday     

On Sums of Products of Bernoulli Variables and Random Permutations

Let {X _k } _k1 be independent Bernoulli random variables with parameters p _k . We study the distribution of the number or runs of length 2: that is . Let S=lim _n S _n . For the particular case p _k =1/(k+B), B being given, we show that the distribution of S is a Beta mixture of Poisson distributions. When B=0 this is a Poisson(1) distribution. For the particular case p _k =p for all k we obtain the generating function of S _n and the limiting distribution of S _n for .     

A Simple and Unusual Bijection for Dyck Paths and its Consequences

In this paper we introduce a new bijection from the set of Dyck paths to itself. This bijection has the property that it maps statistics that appeared recently in the study of patternavoiding permutations into classical statistics on Dyck paths, whose distribution is easy to obtain. We also present a generalization of the bijection, as well as several applications of it to enumeration problems of statistics in restricted permutations.AMS Subject Classification: 05A15, 05A05.     

New families of generating functions for certain class of three-variable polynomials

Recently, Srivastava, Özarslan and Kaanoglu have introduced certain families of three and two variable polynomials, which include Lagrange and Lagrange-Hermite polynomials, and obtained families of two-sided linear generating functions between these families [H.M. Srivastava, M.A. Özarslan, C. Kaanoglu, Some families of generating functions for a certain class of three-variable polynomials, Integr. Transform. Spec. Funct. iFirst (2010) 1-12]. The main object of this investigation is to obtain new two-sided linear generating functions between these families by applying certain hypergeometric transformations. Furthermore, more general families of bilinear, bilateral, multilateral finite series relationships and generating functions are presented for them.     

Some new Chebyshev spaces

In this note we see another circumstance where Chebyshev polynomials play a significant role. In particular, we present some new extended Chebyshev spaces that arise in the asymptotic stability of the zero solution of first order linear delay differential equations with m commensurate delays where a_j,j=0,…,m, are constants and τ>0 is constant.