## Vilenkin Combinatorial Mathematics

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Author: N. Vilenkin

Added by: mirtitles

Added Date: 2015-12-11

Language: English

Subjects: mathematics, combinatorics, series, polynomials, algebra, square roots, polygons, fiboanacic numbers, chess, permutations and combinations, problems

Collections: mir-titles, additional collections

Pages Count: 300

PPI Count: 300

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### Description

In the present book, the aim has been to set forth a variety of combinatorial problems in popular form and understandable language. At the same time, an attempt is made to present some rather involved combinatorial problems and to give the reader an idea of the methods of recurrence relations and generating functions. The first chapter is devoted to the general rules of combinatorics, the rules of sum and product. In the second chapter we investigate permutations and combinations. This traditionally grade-school material is accompanied by an analysis of some amusing examples. In the third chapter, a study is made of combinatorial problems in which certain restrictions are imposed on the combinations. Chapter IV considers problems involving partitions of numbers into integers and- contains a description of certain geometrical methods in combinatorics. Chapter V is devoted to random-walk problems and to a variety of modifications of the arithmetic triangle. Chapter VI takes up recurrence relations, and Chapter VII discusses generating functions and, in particular, the binomial formula. The last section of the book is devoted to combinatorial problems of which there are over 400. This material has been taken from a variety of sources, including Whitworth's Choice and Chance (London, 1901), John Riordan's An Introduction to Combinatorial Analysis (New York, 1958), an interesting book by A. M. Yaglom and I. M. Yaglom entitled Nonelementary Problems in an Elementary Exposition (Moscow, 1954), and various collections of problems given at mathematical Olympiads in the USSR.

This book was translated from the Russian by *George Yankovsky*. The book was published by first Mir Publishers in 1972.

Preface**CHAPTER I.**** THE GENERAL RULES OF COMBINATORICS**

Superstitious cyclists 9

Permutations with repetitions 9

Number systems 10

Secret lock 11

Morse code 11

Wigwag code 11

Electronic digital computer 12

Genetic code 13

General rules of combinatorics 13

Domino problem 15

The crew of a spaceship 15

Checkerboard problems 16

How many people don't know foreign languages? 17

The principle of inclusion and exclusion 18

Where's the mistake? 20

The sieve of Eratosthenes 20

**CHAPTER II. PERMUTATIONS AND COMBINATIONS**

Football championship 22

Permutations without repetitions 22

A science club 22

Permutations of n elements 23

The pr oblem of the rooks 23

Linguistic problems 24

Round dance 25

Permutations with repetitions 25

Anagrams 26

Combinations 27

Genoese lottery 29

Buying cakes 30

Combinations with repetitions 31

The football championship again 32

Properties of combinations 33

A particular case of the principle of inclusion and exclusion 37

Alternating sums of combinations 37

**CHAPTER III. COMBINATORIAL PROBLEMS WITH RESTRICTIONS**

Lions and tigers 39

Building a stairway 39

A bookshelf problem 40

61 61 61 63 64 65 65 67 67 70 71 72 74

King Arthur's Round Table 40

She's got a date 41

A session in telepathy 42

General problem of derangements 44

Subfactorials 45

Caravan in the desert 46

Merry-go-round 47

Standing in line at a ticket office 48

The problem of the two ranks 51

New properties of combinations 51

**CHAPTER IV.**** THE COMBINATORICS OF PARTITIONS**

Dominoes 54

Placing objects into cells 55

A bouquet of flowers 55

The number-of-divisors problem 56

Picking apples 56

Hunting mushrooms 57

Mailing photographs 57

Flags on masts 58

Total number of signals 59

Particle statistics 59

Partitions of integers 59

Mailing packages 60

General problem of postage stamps

Combinatorial problems of information theory

Entrance-exams problem

Paying money

Buying candy

Getting change

Partitioning integers

Arrays of dots

Dual arrays

Euler's formula

**CHAPTER V. COMBINATORICS AND CHESS**

Wandering about town

The arithmetic square

Figurate numbers

The arithmetic triangle

The extended arithmetic triangle

The chess king

The generalized arithmetic triangle 74

Generalized arithmetic triangles and base-m number system 75

Some properties of the numbers C m (k, n) 75

A checker in the corner 77

The arithmetic pentagon 78

Geometric proof of properties of combinations 79

Random walks 80

Brownian motion 81

The queen' s realm 82

Absorbing barriers 83

Random walks on an infinite plane 84

The general problem of the rocks 84

Symmetric arrangements 85

Two knights 87 89 91 91 92 93 94 96 97

**CHAPTER VI. RECURRENCE RELATIONS**

Fibonacci numbers

An alternative proof

The process of successive partitions

Multiplying and dividing numbers

Problems involving polygons

Difficulties of a majordomo

Lucky trolleybus tickets

Recurrence tables

Alternative solution of the majordomo problem

Solution of recurrence relations

Linear recurrence relations with constant coefficients

98

99

100

The case of equal roots of a characteristic equation 102

Application of the theory of recurrence relations to problems of

transmitting information 103

A third solution to the majordomo problem 103

**CHAPTER VII. COMBINATORICS AND SERIES**

Dividing polynomials 104

Algebraic fractions and power series 104

Operations on power series 107

Using power series to prove identities 108

Generating functions 109

Newton's binomial theorem 109

The multinomial theorem 111

Newton's series 112

Extracting square roots 114

Generating functions and recurrence relations 116

Decomposition into partial fractions 116

On a single nonlinear recurrence relation 118

Generating functions and partitions of integers 119

Summary of the combinatorics of partitions 122

Combinatorial Problems 123

Solutions and Answers 152

Index 205