HOW TO COUNT

Enumerative Combinatorics

  1. Basic Principles of Counting
      2. Notation: The number of elements in a set A is denoted #A or |A|.
    1. The one-to-one correspondence principle
    2. The addition principle
    3. The multiplication principle
    4. Counting and probability
  2. Ordered Samples (linearly ordered arrangements)
      3. Strings, finite sequences, words, linear arrangements
    1. Samples with replacement:
      Ordered arrangements with repetition allowed
    2. Samples without replacement:
      Permutations, ordered arrangements without repetitions
  3. Unordered Samples (selections)
    1. Samples without replacement:
      Combinations,subsets, selections without repetitions
    2. Samples with replacement:
      Selections with repetition allowed, multisets
  4. Functions/Distributions
    1. Number of ways to distribute a articles into b boxes
      1. Distinguishable articles, distinguishable boxes
      2. Indistinguishable articles, distinguishable boxes
    2. Number of ways to distribute a articles into b boxes with a given number in each box
      1. Distinguishable articles, distinguishable boxes

        2. Multinomial distribution
        Maxwell-Boltzmann statistics
      2. Indistinguishable articles, distinguishable boxes

        3. Bose-Einstein statistics
        Fermi-Dirac statistics
    3. Extra credit: The Twelve-fold Way
  5. Partitions
    1. Unordered partitions
      Stirling numbers of the second kind
    2. Ordered partitions
  6. Inclusion-exclusion Principle
    1. Derangements
    2. Other applications