Combinatorial Identities for Stirling Numbers
Combinatorial Identities for Stirling Numbers

Author: Jocelyn Quaintance

Publisher: World Scientific

Published: 2015-10-27

Total Pages: 277

ISBN-13: 9814725285

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"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Combinatorial Identities For Stirling Numbers: The Unpublished Notes Of H W Gould
Combinatorial Identities For Stirling Numbers: The Unpublished Notes Of H W Gould

Author: Jocelyn Quaintance

Publisher: World Scientific

Published: 2015-10-27

Total Pages: 277

ISBN-13: 9814725293

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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
Proofs that Really Count

Author: Arthur T. Benjamin

Publisher: American Mathematical Society

Published: 2022-09-21

Total Pages: 210

ISBN-13: 1470472597

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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.

Combinatorics: The Art of Counting
Combinatorics: The Art of Counting

Author: Bruce E. Sagan

Publisher: American Mathematical Soc.

Published: 2020-10-16

Total Pages: 304

ISBN-13: 1470460327

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This book is a gentle introduction to the enumerative part of combinatorics suitable for study at the advanced undergraduate or beginning graduate level. In addition to covering all the standard techniques for counting combinatorial objects, the text contains material from the research literature which has never before appeared in print, such as the use of quotient posets to study the Möbius function and characteristic polynomial of a partially ordered set, or the connection between quasisymmetric functions and pattern avoidance. The book assumes minimal background, and a first course in abstract algebra should suffice. The exposition is very reader friendly: keeping a moderate pace, using lots of examples, emphasizing recurring themes, and frankly expressing the delight the author takes in mathematics in general and combinatorics in particular.

The Art of Proving Binomial Identities
The Art of Proving Binomial Identities

Author: Michael Z. Spivey

Publisher: CRC Press

Published: 2019-05-10

Total Pages: 277

ISBN-13: 1351215809

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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.

James Stirling’s Methodus Differentialis
James Stirling’s Methodus Differentialis

Author: Ian Tweddle

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 301

ISBN-13: 1447100212

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A new translation makes this classic and important text more generally accessible. The text is placed in its contemporary context, but also related to the interests of practising mathematicians today. This book will be of interest to mathematical historians, researchers, and numerical analysts.

Notes On The Binomial Transform: Theory And Table With Appendix On Stirling Transform
Notes On The Binomial Transform: Theory And Table With Appendix On Stirling Transform

Author: Khristo N Boyadzhiev

Publisher: World Scientific

Published: 2018-04-10

Total Pages: 206

ISBN-13: 9813234997

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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.

Bijective Combinatorics
Bijective Combinatorics

Author: Nicholas Loehr

Publisher: CRC Press

Published: 2011-02-10

Total Pages: 600

ISBN-13: 1439848866

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Bijective proofs are some of the most elegant and powerful techniques in all of mathematics. Suitable for readers without prior background in algebra or combinatorics, Bijective Combinatorics presents a general introduction to enumerative and algebraic combinatorics that emphasizes bijective methods.The text systematically develops the mathematical

Handbook of Mathematical Functions
Handbook of Mathematical Functions

Author: Milton Abramowitz

Publisher: Courier Corporation

Published: 1965-01-01

Total Pages: 1068

ISBN-13: 9780486612720

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An extensive summary of mathematical functions that occur in physical and engineering problems

Advanced Combinatorics
Advanced Combinatorics

Author: Louis Comtet

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 353

ISBN-13: 9401021961

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Notwithstanding its title, the reader will not find in this book a systematic account of this huge subject. Certain classical aspects have been passed by, and the true title ought to be "Various questions of elementary combina torial analysis". For instance, we only touch upon the subject of graphs and configurations, but there exists a very extensive and good literature on this subject. For this we refer the reader to the bibliography at the end of the volume. The true beginnings of combinatorial analysis (also called combina tory analysis) coincide with the beginnings of probability theory in the 17th century. For about two centuries it vanished as an autonomous sub ject. But the advance of statistics, with an ever-increasing demand for configurations as well as the advent and development of computers, have, beyond doubt, contributed to reinstating this subject after such a long period of negligence. For a long time the aim of combinatorial analysis was to count the different ways of arranging objects under given circumstances. Hence, many of the traditional problems of analysis or geometry which are con cerned at a certain moment with finite structures, have a combinatorial character. Today, combinatorial analysis is also relevant to problems of existence, estimation and structuration, like all other parts of mathema tics, but exclusively forjinite sets.