A new statistic on Dyck paths for counting 3-dimensional Catalan words

A 3-dimensional Catalan word is a word on three letters so that the subword on any two letters is a Dyck path. For a given Dyck path D, a recently defined statistic counts the number of Catalan words with the property that any subword on two letters is exactly D. In this paper, we enumerate Dyck paths with this statistic equal to certain values, including all primes. The formulas obtained are in terms of Motzkin numbers and Motzkin ballot numbers.     

On certain character sums over p ‐adic rings and their L ‐functions

In this paper we present a new method for evaluating exponential sums associated to a restricted power series in one variable modulo p^l , a power of a prime. We show that for sufficiently large l, these sums can be expressed in terms of Gauss sums. Moreover, we study the associated L ‐functions; we show that they are rational, then we determine their degrees and the weights as Weil numbers of their reciprocal roots and poles. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Potential polynomials and Motzkin paths

A Motzkin path of length n is a lattice path from (0,0) to (n,0) in the plane integer lattice Z×Z consisting of horizontal-steps (1,0), up-steps (1,1), and down-steps (1,−1), which never passes below the x-axis. A u-segment (resp. h-segment) of a Motzkin path is a maximal sequence of consecutive up-steps (resp. horizontal-steps). The present paper studies two kinds of statistics on Motzkin paths: “number of u-segments” and “number of h-segments”. The Lagrange inversion formula is utilized to represent the weighted generating function for the number of Motzkin paths according to the two statistics as a sum of the partial Bell polynomials or the potential polynomials. As an application, a general framework for studying compositions are also provided.     

Hankel determinants of sums of consecutive Motzkin numbers

We use combinatorial methods to evaluate Hankel determinants for the sequence of sums of consecutive t-Motzkin numbers. More specifically, we consider the following determinant:

     

Pattern-restricted permutations composed of 3-cycles

In this paper, we characterize and enumerate pattern-avoiding permutations composed of only 3-cycles. In particular, we answer the question for the six patterns of length 3. We find that the number of permutations composed of n 3-cycles that avoid the pattern 231 (equivalently 312) is given by $3^{n?1}$, while the generating function for the number of those that avoid the pattern 132 (equivalently 213) is given by a formula involving the generating functions for the well-known Motzkin numbers and Catalan numbers. The number of permutations composed of n 3-cycles that avoid the pattern 321 is characterized by a weighted sum involving statistics on Dyck paths of semilength n.     

Peters type polynomials and numbers and their generating functions: Approach with p‐adic integral method

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed.     

Closed-form expression for Hankel determinants of the Narayana polynomials

We considered a Hankel transform evaluation of Narayana and shifted Narayana polynomials. Those polynomials arises from Narayana numbers and have many combinatorial properties. A mainly used tool for the evaluation is the method based on orthogonal polynomials. Furthermore, we provided a Hankel transform evaluation of the linear combination of two consecutive shifted Narayana polynomials, using the same method (based on orthogonal polynomials) and previously obtained moment representation of Narayana and shifted Narayana polynomials.     

A note on 2-distant noncrossing partitions and weighted Motzkin paths

We prove a conjecture of Drake and Kim: the number of 2-distant noncrossing partitions of {1,2,…,n} is equal to the sum of weights of Motzkin paths of length n, where the weight of a Motzkin path is a product of certain fractions involving Fibonacci numbers. We provide two proofs of their conjecture: one uses continued fractions and the other is combinatorial.     

Dyck paths with a first return decomposition constrained by height

We study the enumeration of Dyck paths having a first return decomposition with special properties based on a height constraint. We exhibit new restricted sets of Dyck paths counted by the Motzkin numbers, and we give a constructive bijection between these objects and Motzkin paths. As a byproduct, we provide a generating function for the number of Motzkin paths of height $k$ with a flat (resp. with no flats) at the maximal height.     

Three-term recurrence relations with matrix coefficients. The completely indefinite case

In the space {ol _p ² of vector sequences, we consider the symmetric operatorL generated by the expression (lu)j:=Bj ^uj+1+Aj ^uj+ _B ^j−1/* uj−1, whereu−1 = 0,u ₀,u ₁, … ∈ ℂ ^p ,A _j andB _j arep × p matrices with entries from ℂ,A _j ^* =A_j, and the inversesB _j ⁻¹ (j = 0, 1, …) exist. We state a necessary and sufficient condition for the deficiency numbers of the operatorL to be maximal; this corresponds to the completely indefinite case for the expressionl. Tests for incomplete indefiniteness and complete indefiniteness forl in terms of the coefficientsA _j andB _j are derived. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 709–716, May, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00333.     

Inversive congruential generator over a Galois ring of dimension <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">l</Emphasis></Superscript>

We consider the construction of of an inversive congruential generator over a Galois ring of odd dimension p ^l , whichwas proposed by Solé and Zinoviev for p = 2. Using the estimates of trigonometric sums on the sequences of pseudorandom numbers, we obtain the estimates of a discrepant function, a generated sequence of pseudorandom numbers, and the associated sequence of two-dimensional “overlapping” points.     

Standard Young tableaux and colored Motzkin paths

In this paper, we propose a notion of colored Motzkin paths and establish a bijection between the n-cell standard Young tableaux (SYT) of bounded height and the colored Motzkin paths of length n. This result not only gives a lattice path interpretation of the standard Young tableaux but also reveals an unexpected intrinsic relation between the set of SYTs with at most $2d+1$ rows and the set of SYTs with at most 2d rows.     

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.     

Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs via Stochastic Calculus

We consider two quasi-linear initial-value Cauchy problems on ? ^d : a parabolic system and an hyperbolic one. They both have a first order non-linearity of the form φ(t, x, u)·?u, a forcing term h(t, x, u) and an initial condition u ₀ ∈ L ^∞(? ^d ) ∩ C ^∞(? ^d ), where φ (resp. h) is smooth and locally (resp. globally) Lipschitz in u uniformly in (t, x). We prove the existence of a unique global strong solution for the parabolic system. We show the existence of a unique local strong solution for the hyperbolic one and we give a lower bound regarding its blow up time. In both cases, we do not use weak solution theory but a direct construction based on parabolic schemes studied via a stochastic approach and a regularity result for sequences of parabolic operators. The result on the hyperbolic problem is performed by means of a non-classical vanishing viscosity method.     

Combinatorial numbers in binary recurrences

We give several effective and explicit results concerning the values of some polynomials in binary recurrence sequences. First we provide an effective finiteness theorem for certain combinatorial numbers (binomial coefficients, products of consecutive integers, power sums, alternating power sums) in binary recurrence sequences, under some assumptions. We also give an efficient algorithm (based on genus 1 curves) for determining the values of certain degree 4 polynomials in such sequences. Finally, partly by the help of this algorithm we completely determine all combinatorial numbers of the above type for the small values of the parameter involved in the Fibonacci, Lucas, Pell and associated Pell sequences.      

q‐Bernstein polynomials related to q‐Frobenius–Euler polynomials,l‐functions,and q‐Stirling numbers

The aim of this paper was to derive new identities and relations associated with the q‐Bernstein polynomials, q‐Frobenius–Euler polynomials, l‐functions, and q‐Stirling numbers of the second kind. We also give some applications related to theses polynomials and numbers. Copyright © 2012 John Wiley & Sons, Ltd.     

Condition numbers of Hankel matrices for exponential weights

We obtain the rate of growth of the largest eigenvalues and Euclidean condition numbers of the Hankel matrices for a general class of even exponential weights W²=exp(−2Q) on an interval I. As particular examples, we discuss Q(x)=α|x| on I=R, and Q(x)=(d²−x²)^−α on I=[−d,d].     

A new highly accurate discretization for three‐dimensional singularly perturbed nonlinear elliptic partial differential equations

We employ a new fourth‐order compact finite difference formula based on arithmetic average discretization to solve the three‐dimensional nonlinear singularly perturbed elliptic partial differential equation ε(u_xx + u_yy + u_zz) = f(x, y, z, u, u_x, u_y, u_z), 0 < x, y, z < 1, subject to appropriate Dirichlet boundary conditions prescribed on the boundary, where ε > 0 is a small parameter. We also describe new fourth‐order methods for the estimates of (?u/?x), (?u/?y), and (?u/?z), which are quite often of interest in many physical problems. In all cases, we require only a single computational cell with 19 grid points. The proposed methods are directly applicable to solve singular problems without any modification. We solve three test problems numerically to validate the proposed derived fourth‐order methods. We compare the advantages and implementation of the proposed methods with the standard central difference approximations in the context of basic iterative methods. Numerical examples are given to verify the fourth‐order convergence rate of the methods. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Edge disjoint Hamilton cycles in graphs

Let G be a graph of order n and k ≥ 0 an integer. It is conjectured in [8] that if for any two vertices u and v of a 2(k + 1)‐connected graph G,d _G (u,v) = 2 implies that max{d(u;G), d(v;G)} ≥ (n/2) + 2k, then G has k + 1 edge disjoint Hamilton cycles. This conjecture is true for k = 0, 1 (see cf. [3] and [8]). It will be proved in this paper that the conjecture is true for every integer k ≥ 0. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 8–20, 2000     

The effect of a surface energy term on the distribution of phases in an elastic medium with a two‐well elastic potential

We consider the problem of minimizing among functions u:?^d?Ω→?^d, u_∣?Ω=0, and measurable subsets E of Ω. Here f_h⁺, f^? denote quadratic potentials defined on Ω¯×{symmetric d×d matrices}, h is the minimum energy of f_h⁺ and ε(u) is the symmetric gradient of the displacement field u. An equilibrium state û, Ê of J(u,E) is called one‐phase if E=?? or E=Ω, two‐phase otherwise. For two‐phase states, σ∣?E∩Ω∣ measures the effect of the separating surface, and we investigate the way in which the distribution of phases is affected by the choice of the parameters h??, σ>0. Additional results concern the smoothness of two‐phase equilibrium states and the behaviour of inf J(u,E) in the limit σ↓0. Moreover, we discuss the case of additional volume force potentials, and extend the previous results to non‐zero boundary values. Copyright © 2002 John Wiley & Sons, Ltd.