Издательство D. Reidel Publishing, 1974, -354 pp.
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 combinatorial 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 combinatory analysis) coincide with the beginnings of probability theory in the 17th century. For about two centuries it vanished as an autonomous subject. 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 conceed as 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 mathematics, but exclusively for finite sets.
My idea is here to take the uninitiated reader along a path strewn with particular problems, and I can very well imagine that this jouey may jolt a student who is used to easy generalizations, especially when only some of the questions I treat can be extended at all, and difficult or un- unsolved extensions at that, too. Meanwhile, the treatise remains firmly elementary and almost no mathematics of advanced college level will be necessary.
At the end of each chapter I provide statements in the form of exercises that serve as supplementary material, and I have indicated with a star those that seem most difficult. In this respect, I have attempted to write down these 219 questions with their answers, so they can be consulted as a kind of compendium.
The first items I should quote and recommend from the bibliography are the three great classical treatises of Netto, MacMahon and Riordan. The bibliographical references, all between brackets, indicate the author's name and the year of publication. Thus, [Abel, 1826] refers, in the bibliography of articles, to the paper by Abel, published in 1826. Books are indicated by a star. So, for instance, [Riordan, 1968] refers, in the bibliography of books, to the book by Riordan, published in 1968. Suffixes a, b, c, distinguish, for the same author, different articles that appeared in the same year.
Each chapter is virtually independent of the others, except of the first; but the use of the index will make it easy to consult each part of the book separately.
I have taken the opportunity in this English edition to correct some printing errors and to improve certain points, taking into account the suggestions which several readers kindly communicated to me and to whom I feel indebted and most grateful.
Vocabulary of Combinatorial Analysis.
Partitions of Integers.
Identities and Expansions.
Sieve Formulas.
Stirling Numbers.
Permutations.
Examples оf Inequalities and Estimates.
Fundamental Numerical Tables.
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 combinatorial 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 combinatory analysis) coincide with the beginnings of probability theory in the 17th century. For about two centuries it vanished as an autonomous subject. 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 conceed as 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 mathematics, but exclusively for finite sets.
My idea is here to take the uninitiated reader along a path strewn with particular problems, and I can very well imagine that this jouey may jolt a student who is used to easy generalizations, especially when only some of the questions I treat can be extended at all, and difficult or un- unsolved extensions at that, too. Meanwhile, the treatise remains firmly elementary and almost no mathematics of advanced college level will be necessary.
At the end of each chapter I provide statements in the form of exercises that serve as supplementary material, and I have indicated with a star those that seem most difficult. In this respect, I have attempted to write down these 219 questions with their answers, so they can be consulted as a kind of compendium.
The first items I should quote and recommend from the bibliography are the three great classical treatises of Netto, MacMahon and Riordan. The bibliographical references, all between brackets, indicate the author's name and the year of publication. Thus, [Abel, 1826] refers, in the bibliography of articles, to the paper by Abel, published in 1826. Books are indicated by a star. So, for instance, [Riordan, 1968] refers, in the bibliography of books, to the book by Riordan, published in 1968. Suffixes a, b, c, distinguish, for the same author, different articles that appeared in the same year.
Each chapter is virtually independent of the others, except of the first; but the use of the index will make it easy to consult each part of the book separately.
I have taken the opportunity in this English edition to correct some printing errors and to improve certain points, taking into account the suggestions which several readers kindly communicated to me and to whom I feel indebted and most grateful.
Vocabulary of Combinatorial Analysis.
Partitions of Integers.
Identities and Expansions.
Sieve Formulas.
Stirling Numbers.
Permutations.
Examples оf Inequalities and Estimates.
Fundamental Numerical Tables.