Phu Xuan University Library
|
PXU E-LIB
Sign in
CO

External

Combinatorics

Combinatorics is an upper-level introductory course in enumeration, graph theory, and design theory. The text focuses on combinations and arrangements of discrete structures. There are five major branches of combinatorics that we will touch on in this course: enumeration, graph theory, Ramsey Theory, design theory, and coding theory. We will focus on enumeration, graph theory, and design theory, but will briefly introduce the other two topics.

Combinatorics and Discrete MathematicsMathematics
Author
Joy Morris
Publisher
University of Lethbridge
Source
LibreTexts
License
CC BY-NC-SA
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 103 sections
  1. 01: Introduction
  2. 01: What is Combinatorics
  3. 1.01: Enumeration
  4. 1.02: Graph Theory
  5. 1.03: Ramsey Theory
  6. 1.04: Design Theory
  7. 1.05: Coding Theory
  8. 1.06: Summary
  9. 02: Enumeration
  10. 02: Basic Counting Techniques
  11. 2.01: The Product Rule
  12. 2.02: The Sum Rule
  13. 2.03: Putting Them Together
  14. 2.04: Summing Up
  15. 2.05: Summary
  16. 03: Permutations Combinations and the Binomial Theorem
  17. 3.01: Permutations
  18. 3.02: Combinations
  19. 3.03: The Binomial Theorem
  20. 3.04: Summary
  21. 04: Bijections and Combinatorial Proofs
  22. 4.01: Counting via Bijections
  23. 4.02: Combinatorial Proofs
  24. 4.03: Summary
  25. 05: Counting with Repetitions
  26. 5.01: Unlimited Repetition
  27. 5.02: Sorting a Set that Contains Repetition
  28. 5.03: Summary
  29. 06: Induction and Recursion
  30. 6.01: Recursively-Defined Sequences
  31. 6.02: Basic Induction
  32. 6.03: More Advanced Induction
  33. 6.04: Summary
  34. 07: Generating Functions
  35. 7.01: What is a Generating Function
  36. 7.02: The Generalized Binomial Theorem
  37. 7.03: Using Generating Functions To Count Things
  38. 7.04: Summary
  39. 08: Generating Functions and Recursion
  40. 8.01: Partial Fractions
  41. 8.02: Factoring Polynomials
  42. 8.03: Using Generating Functions to Solve Recursively-Defined Sequences
  43. 8.04: Summary
  44. 09: Some Important Recursively-Defined Sequences
  45. 9.01: Derangements
  46. 9.02: Catalan Numbers
  47. 9.03: Bell Numbers and Exponential Generating Functions
  48. 9.04: Summary
  49. 10: Other Basic Counting Techniques
  50. 10.01: The Pigeonhole Principle
  51. 10.02: Inclusion-Exclusion
  52. 10.03: Summary
  53. 03: Graph Theory
  54. 11: Basics of Graph Theory
  55. 11.01: Background
  56. 11.02: Basic Definitions Terminology and Notation
  57. 11.03: Deletion Complete Graphs and the Handshaking Lemma
  58. 11.04: Graph Isomorphisms
  59. 11.05: Summary
  60. 12: Moving Through Graphs
  61. 12.01: Directed Graphs
  62. 12.02: Walks and Connectedness
  63. 12.03: Paths and Cycles
  64. 12.04: Trees
  65. 12.05: Summary
  66. 13: Euler and Hamilton
  67. 13.01: Euler Tours and Trails
  68. 13.02: Hamilton Paths and Cycles
  69. 13.03: Summary
  70. 14: Graph Coloring
  71. 14.01: Edge Coloring
  72. 14.02: Ramsey Theory
  73. 14.03: Vertex Colouring
  74. 14.04: Summary
  75. 15: Planar Graphs
  76. 15.01: Planar Graphs
  77. 15.02: Eulers Formula
  78. 15.03: Map Colouring
  79. 15.04: Summary
  80. 04: Design Theory
  81. 16: Latin Squares
  82. 16.01: Latin Squares and Sudokus
  83. 16.02: Mutually Orthogonal Latin Squares (MOLS)
  84. 16.03: Systems of Distinct Representatives
  85. 16.04: Summary
  86. 17: Designs
  87. 17.01: Balanced Incomplete Block Designs (BIBD)
  88. 17.02: Constructing Designs and Existence of Designs
  89. 17.03: Fishers Inequality
  90. 17.04: Summary
  91. 18: More Designs
  92. 18.01: Steiner and Kirkman Triple Systems
  93. 18.02: t-Designs
  94. 18.03: Affine Planes
  95. 18.04: Projective Planes
  96. 18.05: Summary
  97. 19: Designs and Codes
  98. 19.01: Introduction
  99. 19.02: Error-Correcting Codes
  100. 19.03: Using the Generator Matrix For Encoding
  101. 19.04: Using the Parity-Check Matrix For Decoding
  102. 19.05: Codes From Designs
  103. 19.06: Summary