Walk Through Combinatorics An Introduction to Enumeration and Graph Theory

ISBN-10: 9814335231
ISBN-13: 9789814335232
Edition: 3rd 2011
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Description: This text provide a complete guide to combinatorics for any introductory courses lasting one or more semesters.

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Book details

List price: $110.00
Edition: 3rd
Copyright year: 2011
Publisher: World Scientific Publishing Co Pte Ltd
Binding: Hardcover
Pages: 568
Size: 6.25" wide x 9.25" long x 1.50" tall
Weight: 2.112
Language: English

This text provide a complete guide to combinatorics for any introductory courses lasting one or more semesters.

Foreword
Preface
Acknowledgments
Basic Methods
Seven Is More Than Six. The Pigeon-Hole Principle
The Basic Pigeon-Hole Principle
The Generalized Pigeon-Hole Principle
Exercises
Supplementary Exercises
Solutions to Exercises
One Step at a Time. The Method of Mathematical Induction
Weak Induction
Strong Induction
Exercises
Supplementary Exercises
Solutions to Exercises
Enumerative Combinatorics
There Are A Lot Of Them. Elementary Counting Problems
Permutations
Strings over a Finite Alphabet
Choice Problems
Exercises
Supplementary Exercises
Solutions to Exercises
No Matter How You Slice It. The Binomial Theorem and Related Identities
The Binomial Theorem
The Multinomial Theorem
When the Exponent Is Not a Positive Integer
Exercises
Supplementary Exercises
Solutions to Exercises
Divide and Conquer. Partitions
Compositions
Set Partitions
Integer Partitions
Exercises
Supplementary Exercises
Solutions to Exercises
Not So Vicious Cycles. Cycles in Permutations
Cycles in Permutations
Permutations with Restricted Cycle Structure
Exercises
Supplementary Exercises
Solutions to Exercises
You Shall Not Overcount. The Sieve
Enumerating The Elements of Intersecting Sets
Applications of the Sieve Formula
Exercises
Supplementary Exercises
Solutions to Exercises
A Function Is Worth Many Numbers. Generating Functions
Ordinary Generating Functions
Recurrence Relations and Generating Functions
Products of Generating Functions
Compositions of Generating Functions
Exponential Generating Functions
Recurrence Relations and Exponential Generating Functions
Products of Exponential Generating Functions
Compositions of Exponential Generating Functions
Exercises
Supplementary Exercises
Solutions to Exercises
Graph Theory
Dots and Lines. The Origins of Graph Theory
The Notion of Graphs. Eulerian Trails
Hamiltonian Cycles
Directed Graphs
The Notion of Isomorphisms
Exercises
Supplementary Exercises
Solutions to Exercises
Staying Connected. Trees
Minimally Connected Graphs
Minimum-weight Spanning Trees. Kruskal's Greedy Algorithm
Graphs and Matrices
Adjacency Matrices of Graphs
The Number of Spanning Trees of a Graph
Exercises
Supplementary Exercises
Solutions to Exercises
Finding A Good Match. Coloring and Matching
Introduction
Bipartite Graphs
Matchings in Bipartite Graphs
More Than Two Color
Matchings in Graphs That Are Not Bipartite
Exercises
Supplementary Exercises
Solutions to Exercises
Do Not Cross. Planar Graphs
Euler's Theorem for Planar Graphs
Polyhedra
Coloring Maps
Exercises
Supplementary Exercises
Solutions to Exercises
Horizons
Does It Clique? Ramsey Theory
Ramsey Theory for Finite Graphs
Generalizations of the Ramsey Theorem
Exercises
Supplementary Exercises
Solutions to Exercises
So Hard To Avoid. Subsequence Conditions on Permutations
Pattern Avoidance
Stack Sortable Permutations
Exercises
Supplementary Exercises
Solutions to Exercises
Who Knows What It Looks Like, But It Exists. The Probabilistic Method
The Notion of Probability
Non-constructive Proofs
Independent Events
The Notion of Independence and Bayes' Theorem
More Than Two Events
Expected Values
Linearity of Expectation
Existence Proofs Using Expectation
Conditional Expectation
Exercises
Supplementary Exercises
Solutions to Exercises
At Least Some Order. Partial Orders and Lattices
The Notion of Partially Ordered Sets
The M�bius Function of a Poset
Lattices
Exercises
Supplementary Exercises
Solutions to Exercises
As Evenly As Possible. Block Designs and Error Correcting Codes
Introduction
Moto-cross Races
Incompatible Computer Programs
Balanced Incomplete Block Designs
New Designs From Old
Existence of Certain BIBDs
A Derived Design of a Projective Plane
Codes and Designs
Coding Theory
Error Correcting Codes
Formal Definitions on Codes
Exercises
Supplementary Exercises
Solutions to Exercises
Are They Really Different? Counting Unlabeled Structures
Enumeration Under Group Action
Introduction
Groups
Permutation Groups
Counting Unlabeled Trees
Counting Rooted Non-plane 1-2 trees
Counting Rooted Non-plane Trees
Counting Unrooted Trees
Exercises
Supplementary Exercises
Solutions to Exercises
The Sooner The Better Combinatorial Algorithms
In Lieu of Definitions
The Halting Problem
Sorting Algorithms
BubbleSort
MergeSort
Comparing the Growth of Functions
Algorithms on Graphs
Minimum-cost Spanning Trees, Revisited
Finding the Shortest Path
Exercises
Supplementary Exercises
Solutions to Exercises
Does Many Mean More Than One? Computational Complexity
Turing Machines
Complexity Classes
The Class P
The Class NP
NP-complete Problems
Other Complexity Classes
Exercises
Supplementary Exercises
Solutions to Exercises
Bibliography
Index

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