scholarly journals Avoidance of classical patterns by Catalan sequences

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 543-558
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Closed Form ◽  
Generating Function ◽  
Final Section ◽  
Catalan Number ◽  
Combinatorial Methods ◽  
General Method

A certain subset of the words of length n over the alphabet of non-negative integers satisfying two restrictions has recently been shown to be enumerated by the Catalan number Cn-1. Members of this subset, which we will denote by W(n), have been termed Catalan words or sequences and are closely associated with the 321-avoiding permutations. Here, we consider the problem of enumerating the members of W(n) satisfying various restrictions concerning the containment of certain prescribed subsequences or patterns. Among our results, we show that the generating function counting the members Of W(n) that avoid certain patterns is always rational for four general classes of patterns. Our proofs also provide a general method of computing the generating function for all the patterns in each of the four classes. Closed form expressions in the case of three-letter patterns follow from our general results in several cases. The remaining cases for patterns of length three, which we consider in the final section, may be done by various algebraic and combinatorial methods.

Coupled dynamics of populations supported by discrete sites and their continuum limit

2010 ◽  
Vol 466 (2121) ◽  
pp. 2561-2586 ◽  
Author(s):  
Timothy R. Field ◽  
Robert J. A. Tough
Keyword(s):  
Evolution Equation ◽  
Closed Form ◽  
Generating Function ◽  
Rate Equation ◽  
Continuum Limit ◽  
Infinite Chain ◽  
Migration Process ◽  
The Mean ◽  
The Continuum ◽  
Birth Death

The illumination of single population behaviour subject to the processes of birth, death and immigration has provided a basis for the discussion of the non-Gaussian statistical and temporal correlation properties of scattered radiation. As a first step towards the modelling of its spatial correlations, we consider the populations supported by an infinite chain of discrete sites, each subject to birth, death and immigration and coupled by migration between adjacent sites. To provide some motivation, and illustrate the techniques we will use, the migration process for a single particle on an infinite chain of sites is introduced and its diffusion dynamics derived. A certain continuum limit is identified and its properties studied via asymptotic analysis. This forms the basis of the multi-particle model of a coupled population subject to single site birth, death and immigration processes, in addition to inter-site migration. A discrete rate equation is formulated and its generating function dynamics derived. This facilitates derivation of the equations of motion for the first- and second-order cumulants, thus generalizing the earlier results of Bailey through the incorporation of immigration at each site. We present a novel matrix formalism operating in the time domain that enables solution of these equations yielding the mean occupancy and inter-site variances in the closed form. The results for the first two moments at a single time are used to derive expressions for the asymptotic time-delayed correlation functions, which relates to Glauber’s analysis of an Ising model. The paper concludes with an analysis of the continuum limit of the birth–death–immigration–migration process in terms of a path integral formalism. The continuum rate equation and evolution equation for the generating function are developed, from which the evolution equation of the mean occupancy is derived, in this limit. Its solution is provided in closed form.

scholarly journals On some combinatorial identities and harmonic sums

2017 ◽  
Vol 13 (07) ◽  
pp. 1695-1709 ◽  
Author(s):  
Necdet Batir
Keyword(s):  
Integral Representation ◽  
Closed Form ◽  
Generating Function ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Generalized Harmonic Numbers

For any [Formula: see text] we first give new proofs for the following well-known combinatorial identities [Formula: see text] and [Formula: see text] and then we produce the generating function and an integral representation for [Formula: see text]. Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that [Formula: see text] and [Formula: see text] where [Formula: see text] are generalized harmonic numbers defined below.

scholarly journals Topological enumeration of complex polynomial vector fields

2014 ◽  
Vol 35 (4) ◽  
pp. 1315-1344 ◽  
Author(s):  
J. TOMASINI
Keyword(s):  
Closed Form ◽  
Orthogonal Group ◽  
Vector Fields ◽  
Main Tool ◽  
Roots Of Unity ◽  
Complex Polynomial ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Appl Math ◽  
General Method

AbstractThe enumeration of combinatorial classes of the complex polynomial vector fields in$ \mathbb{C} $presented by K. Dias [Enumerating combinatorial classes of the complex polynomial vector fields in$ \mathbb{C} $.Ergod. Th. & Dynam. Sys. 33(2013), 416–440] is extended here to a closed form enumeration of combinatorial classes for degree$d$polynomial vector fields up to rotations of the$2(d- 1)\mathrm{th} $roots of unity. The main tool in the proof of this result is based on a general method of enumeration developed by V. A. Liskovets [Reductive enumeration under mutually orthogonal group actions.Acta Appl. Math. 52(1998), 91–120].

DISTRIBUTION FUNCTION FOR SHOT NOISE IN WIGNER-DYSON ENSEMBLES

1996 ◽  
Vol 10 (16) ◽  
pp. 1999-2006 ◽  
Author(s):  
K.A. MUTTALIB ◽  
Y. CHEN
Keyword(s):  
Distribution Function ◽  
Generating Function ◽  
Matrix Model ◽  
Shot Noise ◽  
Random Matrix ◽  
Zero Temperature ◽  
Channel System ◽  
Quantum Regime ◽  
Random Matrix Model ◽  
General Method

We obtain the generating function for shot noise through a random multi-channel system in the zero temperature quantum regime using a random matrix model. The generating function is a product statistic as opposed to a linear one, and we develop a general method to obtain fluctuation properties of such product statistics within a random matrix framework.

On the thermodynamics of nonlinear constitutive relations in gasdynamics

1980 ◽  
Vol 101 (2) ◽  
pp. 225-241 ◽  
Author(s):  
L. C. Woods
Keyword(s):  
Final Section ◽  
Constitutive Relations ◽  
Second Law Of Thermodynamics ◽  
Second Order ◽  
Irreversible Processes ◽  
Burnett Equations ◽  
Necessary Constraints ◽  
Nonlinear Constitutive Relations ◽  
Monatomic Gas ◽  
General Method

The thermodynamics of irreversible processes is normally limited to processes that can be adequately described by linear constitutive relations, like those of Fourier and Newton in a simple gas. In this paper we use thermodynamic arguments to derive the (nonlinear) Burnett equations for a monatomic gas, thus avoiding the complicated kinetic theory by which the equations were discovered and which somewhat obscures the origin of the various terms in the equations. Expressions are given for the entropy, its flux and its production rate correct to second-order in Knudsen number. The theory involves five phenomenological parameters, and as there are eleven coefficients in the second-order terms of Burnett's equations, we are able to deduce several necessary constraints between these coefficients. Compact forms for the equations are found that clarify their physical significance. The general method we have developed is applicable to media other than simple gases.In a final section we use our theory of Burnett's equations to draw some general conclusions concerning the second law of thermodynamics. It is shown that the Clausius-Duhem inequality holds only for the linear theory of constitutive relations; and that axiomatic generalizations of the inequality to nonlinear processes – common in continuum mechanics–fail because the vital distinction between reversible and irreversible processes is not made.

scholarly journals On a New Family of Generalized Stirling and Bell Numbers

10.37236/564 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Toufik Mansour ◽  
Matthias Schork ◽  
Mark Shattuck
Keyword(s):  
Closed Form ◽  
Generating Function ◽  
Recursion Relation ◽  
Stirling Numbers ◽  
Combinatorial Interpretation ◽  
Bell Numbers ◽  
New Family ◽  
Exponential Generating Function ◽  
Rook Numbers ◽  
Generalized Stirling Numbers

A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as $s$-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers, the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore, an analogue of Dobinski's formula is given for these Bell numbers.

A general construction of mutually orthogonal latin squares of prime order

2018 ◽  
Vol 7 (1-2) ◽  
pp. 77-93
Author(s):  
J. A. Saka ◽  
O. O. Oyadare
Keyword(s):  
Structural Properties ◽  
Generating Function ◽  
Generating Functions ◽  
Prime Order ◽  
Latin Squares ◽  
General Construction ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Complete Set ◽  
General Method

This paper presents a general method of constructing a complete set of Mutually Orthogonal Latin Squares (MOLS) of the order of any prime, via the use of generating functions dened on the nite eld of this order. Apart from using the generating function to get a complete set of Mutually Orthogonal Latin Squares, the studies of the generating functions opens up the possibility of getting at the deep structural properties of MOLS. Copious examples were given for detailed illustrations.

Magnetohydrodynamical leaky waves in a layered plasma

1998 ◽  
Vol 59 (1) ◽  
pp. 125-149
Author(s):  
ALEJANDRO G. GONZÁLEZ ◽  
JULIO GRATTON ◽  
SILVIA B. FARINA
Keyword(s):  
Closed Form ◽  
Complex Frequency ◽  
Slab Model ◽  
Leaky Waves ◽  
Numerical Examples ◽  
Physical Mechanisms ◽  
General Method

We investigate the existence of MHD leaky waves in compressible and layered plasmas. We consider perturbations of complex frequency in a slab model with three layers. We develop a general method that includes the ‘cubic modes’ as well as other kinds of leaky waves, and allows us to derive many results in closed form. There are several physical mechanisms that can produce leakage. We give two numerical examples to exhibit the different behaviours that can arise. Finally we comment on the connection of the present results with those obtained by other authors, and correct some errors.

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