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E-raamat: Notes on Counting: An Introduction to Enumerative Combinatorics

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(University of St Andrews, Scotland)
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Notes on Counting: An Introduction to Enumerative CombinatoricsSuurem pilt
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Enumerative combinatorics, in its algebraic and analytic forms, is vital to many areas of mathematics, from model theory to statistical mechanics. This book, which stems from many years' experience of teaching, invites students into the subject and prepares them for more advanced texts. It is suitable as a class text or for individual study. The author provides proofs for many of the theorems to show the range of techniques available, and uses examples to link enumerative combinatorics to other areas of study. The main section of the book introduces the key tools of the subject (generating functions and recurrence relations), which are then used to study the most important combinatorial objects, namely subsets, partitions, and permutations of a set. Later chapters deal with more specialised topics, including permanents, SDRs, group actions and the RedfieldPólya theory of cycle indices, Möbius inversion, the Tutte polynomial, and species.

Arvustused

'It's indeed a very good introduction to enumerative combinatorics and has all the trappings of a pedagogically sound enterprise, in the old-fashioned sense: exercises, good explanations (not too terse, but certainly not too wordy), and mathematically serious (nothing namby-pamby here). It's an excellent book.' Michael Berg, MAA Reviews 'Cameron's Notes on Counting is a clever introductory book on enumerative combinatorics Overall, the text is well-written with a friendly tone and an aesthetic organization, and each chapter contains an ample number of quality exercises. Summing Up: Recommended.' A. Misseldine, CHOICE

Muu info

An introduction to enumerative combinatorics, vital to many areas of mathematics. It is suitable as a class text or for individual study.
Preface ix
1 Introduction
1(11)
1.1 What is counting?
1(6)
1.2 About how many?
7(1)
1.3 How hard is it?
7(1)
1.4 Exercises
8(4)
2 Formal power series
12(17)
2.1 Fibonacci numbers
12(4)
2.2 Formal power series
16(4)
2.3 Variation and generalisation
20(1)
2.4 Relation with analysis
21(2)
2.5 Exponential, logarithmic and binomial series
23(2)
2.6 Exercises
25(4)
3 Subsets, partitions and permutations
29(35)
3.1 Subsets
29(8)
3.2 Partitions
37(5)
3.3 Permutations
42(5)
3.4 Lah numbers
47(1)
3.5 More on formal power series
48(2)
3.6 Unimodality
50(4)
3.7 Appendix: Exponential and logarithm
54(2)
3.8 Exercises
56(8)
4 Recurrence relations
64(42)
4.1 Linear recurrences with constant coefficients
65(12)
4.2 Linear recurrence relations with polynomial coefficients
77(6)
4.3 Some non-linear recurrence relations
83(2)
4.4 Appendix: Euler's Pentagonal Numbers Theorem
85(4)
4.5 Appendix: Some Catalan objects
89(8)
4.6 Appendix: The RSK algorithm
97(2)
4.7 Appendix: Some inverse semigroups
99(2)
4.8 Exercises
101(5)
5 The permanent
106(8)
5.1 Permanents and SDRs
106(1)
5.2 Hall's Theorem
107(1)
5.3 The van der Waerden conjecture
108(2)
5.4 Latin squares
110(3)
5.5 Latin rectangles
113(1)
5.6 Exercises
113(1)
6 q-analogues
114(17)
6.1 Motivation
114(2)
6.2 q-integers
116(2)
6.3 The q-Binomial Theorem
118(1)
6.4 Elementary symmetric functions
119(2)
6.5 Partitions and permutations
121(1)
6.6 Irreducible polynomials
122(2)
6.7 Quantum calculus
124(1)
6.8 Exercises
125(6)
7 Group actions and the cycle Index
131(10)
7.1 Group actions
131(2)
7.2 The Orbit-counting Lemma
133(2)
7.3 The cycle Index
135(2)
7.4 Labelled and unlabelled
137(2)
7.5 Exercises
139(2)
8 Mobius inversion
141(15)
8.1 The Principle of Inclusion and Exclusion
141(3)
8.2 Partially ordered sets
144(2)
8.3 The incidence algebra of a poset
146(1)
8.4 Some Mobius functions
147(2)
8.5 Classical Mobius inversion
149(2)
8.6 The general linear group
151(1)
8.7 Exercises
152(4)
9 The Tutte polynomial
156(14)
9.1 The chromatic polynomial
156(3)
9.2 The Tutte polynomial
159(2)
9.3 Orbit counting and the Tutte polynomial
161(5)
9.4 The Matrix-Tree Theorem
166(2)
9.5 Exercises
168(2)
10 Species
170(17)
10.1 Cayley's Theorem
170(1)
10.2 Species and counting
171(2)
10.3 Examples of species
173(1)
10.4 Operations on species
174(2)
10.5 Cayley's Theorem revisited
176(1)
10.6 What is a species?
177(1)
10.7 Oligomorphic permutation groups
178(3)
10.8 Weights
181(2)
10.9 Exercises
183(4)
11 Analytic methods: a first look
187(8)
11.1 The language of asymptotics
187(1)
11.2 Stirling's formula
188(2)
11.3 Complex analysis
190(2)
11.4 Subadditive and submultiplicative functions
192(2)
11.5 Exercises
194(1)
12 Further topics
195(17)
12.1 Lagrange inversion
195(5)
12.2 Bernoulli numbers
200(4)
12.3 The Euler--Maclaurin sum formula
204(1)
12.4 Poly-Bernoulli numbers
205(2)
12.5 Hayman's Theorem
207(1)
12.6 Theorems of Meir and Moon and of Bender
208(2)
12.7 Exercises
210(2)
13 Bibliography and further directions
212(5)
13.1 The On-line Encyclopedia of Integer Sequences
212(1)
13.2 Books on combinatorial enumeration
213(2)
13.3 Papers cited in the text
215(2)
Index 217
Peter J. Cameron is a Professor in the School of Mathematics and Statistics at the University of St Andrews, Scotland. Much of his work has centred on combinatorics and, since 1992, he has been Chair of the British Combinatorial Committee. He has also worked in group and semigroup theory, model theory, and other subjects such as statistical mechanics and measurement theory. Peter J. Cameron is the recipient of the Senior Whitehead Prize of 2017 from the London Mathematical Society.
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