Combinatorial Identities For Stirling Numbers
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Combinatorial Identities For Stirling Numbers
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Author : Jocelyn Quaintance
language : en
Publisher: World Scientific
Release Date : 2015-10-27
Combinatorial Identities For Stirling Numbers written by Jocelyn Quaintance and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-10-27 with Mathematics categories.
"This book is a unique work which provides an in-depth exploration into the mathematical expertise, philosophy, and knowledge of H W Gould. It is written in a style that is accessible to the reader with basic mathematical knowledge, and yet contains material that will be of interest to the specialist in enumerative combinatorics. This book begins with exposition on the combinatorial and algebraic techniques that Professor Gould uses for proving binomial identities. These techniques are then applied to develop formulas which relate Stirling numbers of the second kind to Stirling numbers of the first kind. Professor Gould's techniques also provide connections between both types of Stirling numbers and Bernoulli numbers. Professor Gould believes his research success comes from his intuition on how to discover combinatorial identities. This book will appeal to a wide audience and may be used either as lecture notes for a beginning graduate level combinatorics class, or as a research supplement for the specialist in enumerative combinatorics."--
Combinatorial Identities
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Author : John Riordan
language : en
Publisher:
Release Date : 1979
Combinatorial Identities written by John Riordan and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1979 with Mathematics categories.
Commutation Relations Normal Ordering And Stirling Numbers
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Author : Toufik Mansour
language : en
Publisher: CRC Press
Release Date : 2015-09-18
Commutation Relations Normal Ordering And Stirling Numbers written by Toufik Mansour and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-09-18 with Mathematics categories.
Commutation Relations, Normal Ordering, and Stirling Numbers provides an introduction to the combinatorial aspects of normal ordering in the Weyl algebra and some of its close relatives. The Weyl algebra is the algebra generated by two letters U and V subject to the commutation relation UV - VU = I. It is a classical result that normal ordering pow
Stirling Numbers
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Author : Elena Deza
language : en
Publisher: World Scientific
Release Date : 2023-12-27
Stirling Numbers written by Elena Deza and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-12-27 with Mathematics categories.
Stirling numbers are one of the most known classes of special numbers in Mathematics, especially in Combinatorics and Algebra. They were introduced by Scottish mathematician James Stirling (1692-1770) in his most important work, Differential Method with a Tract on Summation and Interpolation of Infinite Series (1730). Stirling numbers have a rich history; many arithmetic, number-theoretical, analytical and combinatorial connections; numerous classical properties; as well as many modern applications.This book collects much of the scattered material on the two subclasses of Stirling numbers to provide a holistic overview of the topic. From the combinatorial point of view, Stirling numbers of the second kind, S(n, k), count the number of ways to partition a set of n different objects (i.e., a given n-set) into k non-empty subsets. Stirling numbers of the first kind, s(n, k), give the number of permutations of n elements with k disjoint cycles. Both subclasses of Stirling numbers play an important role in Algebra: they form the coefficients, connecting well-known sets of polynomials.This book is suitable for students and professionals, providing a broad perspective of the theory of this class of special numbers, and many generalisations and relatives of Stirling numbers, including Bell numbers and Lah numbers. Throughout the book, readers are provided exercises to test and cement their understanding.
Combinatorial Identities For Stirling Numbers The Unpublished Notes Of H W Gould
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Author : Jocelyn Quaintance
language : en
Publisher: World Scientific
Release Date : 2015-10-27
Combinatorial Identities For Stirling Numbers The Unpublished Notes Of H W Gould written by Jocelyn Quaintance and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-10-27 with Mathematics categories.
This book is a unique work which provides an in-depth exploration into the mathematical expertise, philosophy, and knowledge of H W Gould. It is written in a style that is accessible to the reader with basic mathematical knowledge, and yet contains material that will be of interest to the specialist in enumerative combinatorics. This book begins with exposition on the combinatorial and algebraic techniques that Professor Gould uses for proving binomial identities. These techniques are then applied to develop formulas which relate Stirling numbers of the second kind to Stirling numbers of the first kind. Professor Gould's techniques also provide connections between both types of Stirling numbers and Bernoulli numbers. Professor Gould believes his research success comes from his intuition on how to discover combinatorial identities.This book will appeal to a wide audience and may be used either as lecture notes for a beginning graduate level combinatorics class, or as a research supplement for the specialist in enumerative combinatorics.
Proofs That Really Count
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Author : Arthur T. Benjamin
language : en
Publisher: American Mathematical Society
Release Date : 2022-09-21
Proofs That Really Count written by Arthur T. Benjamin and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-09-21 with Mathematics categories.
Mathematics is the science of patterns, and mathematicians attempt to understand these patterns and discover new ones using a variety of tools. In Proofs That Really Count, award-winning math professors Arthur Benjamin and Jennifer Quinn demonstrate that many number patterns, even very complex ones, can be understood by simple counting arguments. The book emphasizes numbers that are often not thought of as numbers that count: Fibonacci Numbers, Lucas Numbers, Continued Fractions, and Harmonic Numbers, to name a few. Numerous hints and references are given for all chapter exercises and many chapters end with a list of identities in need of combinatorial proof. The extensive appendix of identities will be a valuable resource. This book should appeal to readers of all levels, from high school math students to professional mathematicians.
Notes On The Binomial Transform Theory And Table With Appendix On Stirling Transform
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Author : Boyadzhiev Khristo N
language : en
Publisher: World Scientific
Release Date : 2018-04-10
Notes On The Binomial Transform Theory And Table With Appendix On Stirling Transform written by Boyadzhiev Khristo N and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-04-10 with Mathematics categories.
The binomial transform is a discrete transformation of one sequence into another with many interesting applications in combinatorics and analysis. This volume is helpful to researchers interested in enumerative combinatorics, special numbers, and classical analysis. A valuable reference, it can also be used as lecture notes for a course in binomial identities, binomial transforms and Euler series transformations. The binomial transform leads to various combinatorial and analytical identities involving binomial coefficients. In particular, we present here new binomial identities for Bernoulli, Fibonacci, and harmonic numbers. Many interesting identities can be written as binomial transforms and vice versa. The volume consists of two parts. In the first part, we present the theory of the binomial transform for sequences with a sufficient prerequisite of classical numbers and polynomials. The first part provides theorems and tools which help to compute binomial transforms of different sequences and also to generate new binomial identities from the old. These theoretical tools (formulas and theorems) can also be used for summation of series and various numerical computations. In the second part, we have compiled a list of binomial transform formulas for easy reference. In the Appendix, we present the definition of the Stirling sequence transform and a short table of transformation formulas. Contents: Theory of the Binomial Transform: Introduction Prerequisite: Special Numbers and Polynomials Euler's Transformation for Series Melzak's Formula and Related Formulas Special Properties. Creating New Identities Binomial Transforms of Products Special Formulas and Power Series with Binomial Sums Table of Binomial Transforms: Assorted Binomial Formulas Identities Involving Harmonic Numbers Transforms of Binomial Coefficients Transforms of Special Numbers and Polynomials Transforms of Trigonometric and Hyperbolic Functions and Applications to Some Trigonometric Integrals Transforms of Some Special Functions Appendix: The Stirling Transform of Sequences Readership: Graduate and researchers in the areas of number theory, discrete mathematics, combinatorics, statistics working with applications using the binomial transform. Keywords: Binomial Coefficients;Binomial Identities;Binomial Sums;Binomial Transform;Euler's Series Transformation;Discrete Mathematics;Finite Differences;Stirling Numbers of the First Kind;Stirling Numbers of the Second Kind;Stirling Transform;Special Numbers and Polynomials;Harmonic Numbers;Bernoulli Numbers;Fibonacci Numbers;Melzak's Formula;Exponential Polynomials;Geometric Polynomials;Laguerre Polynomials;Trigonometric IntegralsReview: Key Features: This is the first, long-overdue book on the subject. (At present, there are no competing books) The book provides interesting new material for researchers in discrete mathematics and will serve as a valuable reference for binomial identities, binomial transform formulas, and Euler series transformations
Combinatorics
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Author : Derek Allan Holton
language : en
Publisher:
Release Date : 1990
Combinatorics written by Derek Allan Holton and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990 with Combinatorial analysis categories.
Principles And Techniques In Combinatorics
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Author : Chuan-Chong Chen
language : en
Publisher: World Scientific Publishing Company Incorporated
Release Date : 1992
Principles And Techniques In Combinatorics written by Chuan-Chong Chen and has been published by World Scientific Publishing Company Incorporated this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with Combinations categories.
1. Permutations and combinations. 1.1. Two basic counting principles -- 1.2. Permutations -- 1.3. Circular permutations -- 1.4. Combinations -- 1.5. The injection and bijection principles -- 1.6. Arrangements and selections with repititions -- 1.7. Distribution problems -- Exercise 1 -- 2. Binomial coefficients and multinomial coefficients. 2.1. Introduction -- 2.2. The Binomial Theorem -- 2.3. Combinatorial identities -- 2.4. The Pascal's Triangle -- 2.5. Chu Shih-Chieh's Identity -- 2.6. Shortest routes in a rectangular grid -- 2.7. Some properties of Binomial coefficients -- 2.8. Multinomial coefficients and the Multinomial Theorem -- Exercise 2 -- 3. The Pigeonhole Principle and Ramsey Numbers. 3.1. Introduction -- 3.2. The Pigeonhole Principle -- 3.3. More examples -- 3.4. Ramsey type problems and Ramsey numbers -- 3.5. Bounds for Ramsey numbers -- Exercise 3 -- 4. The Principle of Inclusion and Exclusion. 4.1. Introduction -- 4.2. The principle -- 4.3. A generalization -- 4.4. Integer solutions and shortest routes -- 4.5. Subjective mappings and Stirling numbers of the second kind -- 4.6. Derangements and a generalization -- 4.7. The Sieve of Eratosthenes and Euler [symbol]-function -- 4.8. The "Probleme des Menages" -- Exercise 4 -- 5. Generating functions. 5.1. Ordinary generating functions -- 5.2. Some modelling problems -- 5.3. Partitions of integers -- 5.4. Exponential generating functions -- Exercise 5 -- 6. Recurrence relations. 6.1. Introduction -- 6.2. Two examples -- 6.3. Linear homogeneous recurrence relations -- 6.4. General linear recurrence relations -- 6.5. Two applications -- 6.6. A system of linear recurrence relations -- 6.7. The method of generating functions -- 6.8. A nonlinear recurrence relation and Catalan numbers -- 6.9. Oscillating permutations and an exponential generating function -- Exercise 6
The Art Of Proving Binomial Identities
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Author : Michael Z. Spivey
language : en
Publisher: CRC Press
Release Date : 2019-05-10
The Art Of Proving Binomial Identities written by Michael Z. Spivey and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-05-10 with Mathematics categories.
The book has two goals: (1) Provide a unified treatment of the binomial coefficients, and (2) Bring together much of the undergraduate mathematics curriculum via one theme (the binomial coefficients). The binomial coefficients arise in a variety of areas of mathematics: combinatorics, of course, but also basic algebra (binomial theorem), infinite series (Newton’s binomial series), differentiation (Leibniz’s generalized product rule), special functions (the beta and gamma functions), probability, statistics, number theory, finite difference calculus, algorithm analysis, and even statistical mechanics.