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ستاره غیر فعالستاره غیر فعالستاره غیر فعالستاره غیر فعالستاره غیر فعال
 

اثر: Peter J. Cameron

تعداد صفحه: 227

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توضیحات (خلاصه پیشگفتار):

Combinatorics is the science of pattern and arrangement. A typical problem in combinatorics asks whether it is possible to arrange a collection of objects according to certain rules. If the arrangement is possible, the next question is a counting question: how many different arrangements are there? This is the topic of the present book.

Often a small change in the detail of a problem turns an easy question into one which appears impossibly difficult.
• In how many ways can the numbers 1, . . . ,n be placed in the cells of an n×n grid, with no restriction on how many times each is used? Since each of the n2 cells can have its entry chosen independently from a set of n possibilities, the answer is n(NumberOfCells).

• In how many ways can the arrangement be made if each number must occur once in each row? Once we notice that each row must be a permutation of the numbers 1, . . . ,n, and that the permutations can be chosen independently, we see that the answer is (n!)n (as there are n! permutations of the numbers 1, . . . ,n).
• In how many ways can the arrangement be made if each number must occur once in each row and once in each column? For this problem, there is no formula for the answer. Such an arrangement is called a Latin square. The number of Latin squares with n up to 11 has been found by brute-force calculation. For larger values, we don’t even have good estimates: the best known upper and lower bounds differ by a factor which is exponentially large in terms of the number of cells.

Not all problems are as hard as this. In this book you will learn how to count the number of permutations which move every symbol, strings of zeros and ones containing no occurrence of a fixed substring, invertible matrices of given size over a finite field, expressions for a given positive integer as a sum of positive integers (the answers are different depending on whether we care about the order of the summands or not), trees and graphs on a given set of vertices, orbits of the general linear group on tuples of vectors, zero-one matrices of unspecified size with a given number of ones, any many more.

Very often I will not be content with giving you one proof of a theorem. I may return to an earlier result armed with a new technique and give a totally different proof. We learn something from having several proofs of the same result. In particular, there are two quite different styles of proof for results in enumerative combinatorics. Consider a result which asserts that two counting functions F(n) and G(n) are equal. We might prove this by showing that their generating functions are equal, perhaps using analytic techniques of some kind. Alternatively, we might prove the result by finding a bijection between the sets of objects counted by F(n) and G(n).

I should stress, though, that the book is not full of big theorems. Tim Gowers, in a perceptive article on “The two cultures of mathematics”, distinguishes branches of mathematics in which theorems are all-important from those where the emphasis is on techniques; enumerative combinatorics falls on the side of techniques. (In the past this has led to some disparagement of combinatorics by other mathematicians. Many people know that Henry Whitehead said “Combinatorics is the slums of topology”. A more honest appraisal is that the techniques of combinatorics pervade all of mathematics, even the most theorem-rich parts.)

The notes which became this book were for a course on Enumerative and Asymptotic Combinatorics at Queen Mary, University of London, in the spring of 2003. Since then, the reference material for the subject has been greatly expanded by the publication of Richard Stanley’s two-volume work on Enumerative Combinatorics, as well as the Web book by Flajolet and Sedgwick. (References to these and many others can be found in the bibliography at the end.) Many of these are encyclopaedic in nature. I hope that this book will be an introduction to the subject, which will encourage you to look further and to tackle some of the weightier tomes.

What do you need to know to read this book? You should have had some exposure to basic topics in undergraduate mathematics:

• Real and complex analysis (limits, convergence, power series, Cauchy’s theorem, singularities of complex functions);

• Algebra (groups and rings, group actions, Smith normal form);

• Combinatorics (though this is less essential; it is a subject that you can learn as you go along).

I am grateful to Morteza Mohammed-Noori, who used my course notes for a course of his own in Tehran, and did a very thorough proof-reading job.

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