Combinatorics - 1st Edition - ISBN: 9780127219400, 9781483266114


1st Edition

Authors: N. Ya. Vilenkin
eBook ISBN: 9781483266114
Imprint: Academic Press
Published Date: 28th January 1971
Page Count: 314
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Combinatorics deals with simple combinatorial problems, recurrence relations, and generating functions, particularly the binomial expansions. The book expounds on the general rules of combinatorics, the rule of sum, the rule of product, samples, permutations, combinations, and arrangements of subjects with various restrictions. The text also explains ordered or unordered partitions of numbers, geometric methods, random walk problems, and variants of the arithmetical triangle. One example of the use of combinatorics is the choice of the number 3 in the genetic code. Another example involves the choice of crew for a spaceship where it is necessary to consider the psychological conditions of the applicants for space travel. The text also investigates the sieve of Erastothenes whose problem concerns finding all the primes in the sequence of natural numbers from 1 to N. The book also tackles the application of power series to proof of identities, the binomial series expansion, decomposition into elementary fractions, and nonlinear recurrence relation. The book can be highly educational and interesting to students or academicians involved in mathematics, algebra, and statistics.

Table of Contents


Preface to the Russian Edition

I. General Rules of Combinatorics

Superstitious Cyclists

Samples with Repetitions

Systems of Numeration

Combination Lock

The Morse Code

Signaling at Sea

Electronic Computers

Genetic Code

General Rules of Combinatorics

A Domino Problem

Crew of a Spaceship

A Checkers Problem

How Many People Do Not Know a Foreign Language?

The Principle of Inclusion and Exclusion

Find the Error!

The Sieve of Erastothenes

II. Samples, Permutations, and Combinations

Soccer Tournament

Samples without Repetitions

Learned Society


A Rook Problem

Linguistic Problems

Dancing in a Circle

Permutations with Repetitions



The Genoa Lottery

Buying Pastry

Combinations with Repetitions

The Soccer Tournament Revisited

Properties of Combinations

Special Case of the Inclusion and Exclusion Formula

Alternating Sums of Combinations

III. Combinatorial Problems with Restrictions

Lions and Tigers

Construction of a Staircase

A Bookshelf

Knights of the Round Table

A Girl Has a Date

A Telepathic Séance



Caravan in the Desert

On the Merry-Go-Round

Queue at the Box Office

Problem of Two Rows

Additional Properties of Combinations

IV. Distributions and Partitions

A Domino Game

Putting Things in Boxes

A Bouquet

The Number of Divisors

Picking Apples

Picking Mushrooms

Sending Photographs

Flags on Masts

Count of All Signals

Various Statistics

Partitions of Numbers

Mailing of Printed Matter

The General Postage Problem

Combinatorial Problems in Information Theory

Problem of the University Applicant

Paying for a Purchase

Purchase of Candy

Getting Change

(Unordered) Partitions of an Integer

Use of Graphs

Dual Graphs

Euler's Formula

V. Combinatorics on a Chessboard

Walking in a Town

The Arithmetical Square

Figurate Numbers

Arithmetical Triangle

The Extended Arithmetical Triangle

Chess King

The Generalized Arithmetical Triangle

Generalized Arithmetical Triangles and Numbers to the Base m

Some Properties of the Numbers Cm(k, n)

Checker in a Corner

The Arithmetical Pentagon

Geometric Proofs of Properties of Combinations

Random Walk

Brownian Motion

In the Realm of the Tsarina of Shemakhan

An Absorbing Wall

Roaming on an Infinite Plane

The General Rook Problem

Symmetric Arrangements

Two Knights

VI. Recurrence Relations

Fibonacci Numbers

Another Method of Proof

Successive Partitions

Multiplication and Division of Numbers

Polygon Problems

The Head Steward Faces a Difficulty

Lucky Bus Ticket Numbers

Recurrent Tables

Another Solution of the Problem of the Chief Steward

Solution of Recurrence Relations

Linear Recurrence Relations with Constant Coefficients

The Case of Equal Roots of the Characteristic Equation

Application of the Theory of Recurrence Relations to Information Theory

Third Solution of the Problem of the Chief Steward

VII. Combinatorics and Series

Division of Polynomials

Algebraic Fractions and Power Series

Operations on Power Series

Application of Power Series to Proofs of Identities

Generating Functions

The Binomial Expansion

The Multinomial Expansion

The Binomial Series Expansion

Computation of Square Roots

Generating Functions and Recurrence Relations

Decomposition into Elementary Fractions (Partial Fraction Decomposition)

On a Certain Nonlinear Recurrence Relation

Generating Functions and Partitions of Numbers

Summary of Results on Combinatorial Distributions

VIII. Problems in Combinatorics

Answers and Hints



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© Academic Press 1971
Academic Press
eBook ISBN:

About the Author

N. Ya. Vilenkin