导语

  

内容提要

  

    本书主要包括计数组合学、鸽巢原理、几何图形数、加性和乘性原则、排列和组合、组合模形、循环排列和乱序、二项级数和生成函数、牛顿二项级数、递归关系、斐波那契递归关系、调和数、伯努利数、卡塔兰数、线性空间和递归序列、计数和对称性、代数发现、伯恩赛德引理等内容。计数组合学是数学中一些最有趣的问题的来源。这些问题通常可以通过巧妙的创造性的观察来解决，我们称之为组合推理，在本书的所有描述、示例和问题中都强调了这种思维。组合学在计算机科学、概率统计和离散优化等领域有许多重要的应用。

PREFACE
PART I  THE BASICS OF ENUMERATIVE COMBINATORICS
  1    Initial EnCOUNTers with Combinatorial Reasoning
    1.1  Introduction
    1.2  The Pigeonhole Principle
    1.3  Tiling Chessboards with Dominoes
    1.4  Figurate Numbers
    1.5  Counting Tilings of Rectangles
    1.6  Addition and Multiplication Principles
    1.7  Summary and Additional Problems, 46 References
  2    Selections, Arrangements, and Distributions
    2.1  Introduction
    2.2  Permutations and Combinations
    2.3  Combinatorial Models
    2.4  Permutations and Combinations with Repetitions
    2.5  Distributions to Distinct Recipients
    2.6  Circular Permutations and Derangements
    2.7  Summary and Additional Problems
    Reference
  3  Binomial Series and Generating Functions
    3.1  Introduction
    3.2  The Binomial and Multinomial Theorems
    3.3  Newton's Binomial Series
    3.4  Ordinary Generating Functions
    3.5  Exponential Generating Functions
    3.6  Summary and Additional Problems
    References
  4   Alternating Sums, Inclusion-Exclusion Principle, Rook
    Polynomials, and Fibonacci Nim
    4.1  Introduction
    4.2  Evaluating Alternating Sums with the DIE Method
    4.3  The Principle of Inclusion-Exclusion (PIE)
    4.4  Rook Polynomials
    4.5  (Optional) ZeckendorfRepresentations  and Fibonacci Nim
    4.6  Summary and Additional Problems
    References, 2 l
  5  Recurrence Relations
    5,l  Introduction
    5.2  The Fibonacci Recurrence Relation
    5.3  Second-Order Recurrence Relations
    5.4  Higher-Order Linear Homogeneous Recurrence  Relations
    5.5  Nonhomogeneous Recurrence Relations
    5.6  Recurrence Relations and Generating Functions
    5.7  Summary and Additional Problems
    References
  6  Special Numbers
    6.1  Introduction
    6.2  Stirling Numbers
    6.3  Harmonic Numbers
    6.4  Bernoulli Numbers
    6.5  Eulerian Numbers
    6.6  Partition Numbers
    6.7  Catalan Numbers
    6.8  Summary and Additional Problems
    References
PART II  TWO ADDITIONAL TOPICS IN ENUMERATION
  7   Linear Spaces and Recurrence Sequences
    7.1  Introduction
    7.2  Vector Spaces of Sequences
    7.3  Nonhomogeneous Recurrences and Systems  of Recurrences
    7.4  Identities for Recurrence Sequences
    7.5  Summary and Additional Problems
  8   Counting with Symmetries
    8.1  Introduction
    8.2  Algebraic Discoveries
    8.3  Burnside's Lemma
    8.4  The Cycle Index and P61ya's Method of Enumeration
    8.5  Summary and Additional Problems
    References
PART III  NOTATIONS INDEX, APPENDICES, AND SOLUTIONS   TO SELECTED ODD PROBLEMS
  Index of Notations
  Appendix A  Mathematical Induction
    A.1 Principle of Mathematical Induction
    A.2 Principle of Strong Induction
    A.3 Well Ordering Principle
  Appendix B  Searching the Online Encyclopedia of lnteger Sequences (OEIS)
    B.1  Searching a Sequence
    B.2 Searching an Array
    B.3 Other Searches
    B.4 Beginnings of OEIS
Appendix C  Generalized Vandermonde Determinants
  Hints, Short Answers, and Complete Solutions to Selected
  Odd Problems
INDEX