HOW TO COUNT without counting:     Enumerative Combinatorics

  1. Basic principles of counting
      2. Notation: The number of elements in a set A is denoted #A or |A| or n(A).
    1. The one-to-one correspondence principle
      1. If A <--> B, then |A| = |B|.
      2. Tree diagrams: Possibilities <--> paths to leaves <--> leaves.
    2. The addition principle: counting by cases
      1. Disjoint sets/cases
      2. Subtraction formula
          | Ac | = |U\A| = |U| - |A| if U = universal set.
      3. Inclusion-Exclusion Principle
      4. Tree diagrams: #leaves = sum of #leaves in subtrees.
    3. The multiplication principle: counting by stages
    4. Exponential formulas: counting functions, 1-1 functions, subsets
      1. # functions f: A --> B is |B||A|.
      2. # 1-1 functions f: A --> B is |B||A|.
      3. # subsets of set A is 2|A|.
    5. Division formulas
      1. # distinguishable or unordered cases from # distinguishable or ordered cases
          Example: Combinations
      2. Counting and probability: P(A) = |A|/|S|.
    6. Forms for answers
      1. Explicit, closed-form formulas
      2. Recurrences
      3. Generating functions
    7. Key questions
      1. What would a list of the possibilities look like?
      2. Is order important?
      3. What cases are indistinguishable?
      4. What are the constraints on repetitions?
      5. What is the answer in small cases?
  2. Linearly ordered arrangements
      3. Finite sequences, strings, words, ...
    1. Ordered arrangements without repetitions, permutations
    2. Ordered arrangements with unlimited repetition allowed
    3. Ordered arrangements with limited repetition allowed
      1. Specified numbers of repetitions: permutations of a multiset
      2. Bounded numbers of repetitions: r-permutations of a multiset
      3. Other restrictions
  3. Unordered selections
    1. Selections without repetitions
      Combinations,subsets, samples without replacement
    2. Selections with unlimited repetition allowed
      Multisets, samples with replacement, nonnegative integer solutions of x1 + ... + xn = r.
    3. Selections with limited repetition allowed
      Submultisets, r-combinations of a multiset with n distinct elements
  4. Sampling
  5. Distributions/functions
    1. Number of ways to distribute a articles into b boxes
      1. Distinguishable articles, distinguishable boxes, unordered distribution
      2. Distinguishable articles, distinguishable boxes, ordered distribution--flagpole problem
      3. Indistinguishable articles, distinguishable boxes
          Flagpole problem with indistinguishable flags
    2. Number of ways to distribute a articles into b boxes with a given number in each box
      1. Distinguishable articles, distinguishable boxes
        1. Multinomial distribution
        2. Maxwell-Boltzmann statistics
      2. Indistinguishable articles, distinguishable boxes
        1. Bose-Einstein statistics
        2. Fermi-Dirac statistics
    3. Number of ways to distribute a articles into b indistinguishable boxes
      1. Distinguishable articles: partitions of a set
        1. Stirling numbers of the second kind
        2. Bell numbers
      2. Indistinguishable articles: partitions of an integer
    4. The Twelvefold Way   (ps) (pdf)
      The Sixteenfold Way   (ps) (pdf)
  6. Generating functions
    1. Ordinary generating functions
    2. Exponential generating functions