Enumeration problems associated with Donaghey’s transformation

2019 ◽  
pp. 5-25
Author(s):  
Viktor Byzov
Keyword(s):  
Generating Functions ◽  
Enumeration Problems

In this paper we consider enumeration problems associated with Donaghey’s transformation. We discuss two groups of questions. The first one is related to the enumeration of fragments of transformation orbits, which are referred to as the “arcs”. The second group of questions is concerned with finding the number of vertices in rotation graphs — a specific family of graphs that is by nature an approximation of Donaghey’s transformation. The basic results of this work are formulated in the form of generating functions and corresponding asymptotics.

scholarly journals Multivariate Asymptotics for Products of Large Powers with Applications to Lagrange Inversion

10.37236/1440 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Edward A. Bender ◽  
L. Bruce Richmond
Keyword(s):  
Limit Theorem ◽  
Generating Functions ◽  
Asymptotic Estimate ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Combinatorial Enumeration ◽  
Lagrange Inversion ◽  
Enumeration Problems

An asymptotic estimate is given for the coefficients of products of large powers of generating functions. This theorem and another local limit theorem which is useful for conditioning are applied to various combinatorial enumeration problems that involve multivariate Lagrange inversion.

Enumeration of Graphs with Given Partition

1968 ◽  
Vol 20 ◽  
pp. 40-47 ◽  
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Generating Functions ◽  
Review Article ◽  
Polya's Theorem ◽  
Enumeration Problems ◽  
Pólya’S Theorem ◽  
General Graphs

In this paper we use a generalized form of Polya's theorem (1) to obtain generating functions for the number of ordinary graphs with given partition and for the number of bicoloured graphs with given bipartition. Both the points and lines of the graphs are taken as unlabelled. These graph enumeration problems were proposed by Harary in his review article (4). Read (7, 8) solved the problem for unlabelled general graphs and labelled ordinary graphs.

Decomposition Based Generating Functions for Sequences

1977 ◽  
Vol 29 (5) ◽  
pp. 971-1009 ◽  
Author(s):  
D. M. Jackson ◽  
R. Aleliunas
Keyword(s):  
Generating Functions ◽  
Combinatorial Enumeration ◽  
Enumeration Problems ◽  
Planar Maps ◽  
Equivalent Problems

Numerous combinatorial enumeration problems may be reduced to equivalent problems of enumerating sequences with prescribed restrictions. For example, the expression, given by Tutte [38], for the number of planar maps may be derived (see Cori and Richard [12]) by essentially a sequence enumeration technique. The correspondence between a set of configurations which are to be enumerated and an appropriate set of sequences is often complicated. Indeed, the existence of such a correspondence has occasionally only been discovered fortuitously by observing the equality of two counting series (see, for example, Klarner [25]).

MORE ON ADDITIVE ENUMERATION PROBLEMS OVER TREES

1990 ◽  
Vol 10 (4) ◽  
pp. 396-401
Author(s):  
Zhenyu Wang ◽  
Chaoyi Sun
Keyword(s):  
Enumeration Problems

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
Author(s):  
Eda Yuluklu ◽  
Yilmaz Simsek ◽  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Special Polynomials ◽  
Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

Symplectic capacities

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Euclidean Space ◽  
Generating Functions ◽  
Weinstein Conjecture ◽  
Finite Dimensional ◽  
Hofer Metric

This chapter returns to the problems which were formulated in Chapter 1, namely the Weinstein conjecture, the nonsqueezing theorem, and symplectic rigidity. These questions are all related to the existence and properties of symplectic capacities. The chapter begins by discussing some of the consequences which follow from the existence of capacities. In particular, it establishes symplectic rigidity and discusses the relation between capacities and the Hofer metric on the group of Hamiltonian symplectomorphisms. The chapter then introduces the Hofer–Zehnder capacity, and shows that its existence gives rise to a proof of the Weinstein conjecture for hypersurfaces of Euclidean space. The last section contains a proof that the Hofer–Zehnder capacity satisfies the required axioms. This proof translates the Hofer–Zehnder variational argument into the setting of (finite-dimensional) generating functions.

Poisson Brackets & Angular Momentum

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Generating Functions ◽  
First Integrals ◽  
Lagrangian Mechanics ◽  
Poisson Brackets ◽  
Coordinate Transformations ◽  
Canonical Transformations ◽  
Gauge Transformations ◽  
The Third ◽  
Point Transformations ◽  
Canonical Coordinate

This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

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