the "balls and urns" model

• general overview: [1]
• The "Twelvefold way": [1] [2] [3]

  balls per urn:	unrestricted	≤1	≥1
  n labelled balls, m labelled urns	n-tuples of m things	n-permutations of m things	partitions of {1, ... , n} into ≤ m ordered parts
  n unlabelled balls, m labelled urns	n-multicombinations of m things	n-multicombinations of m things	n-combinations of m things
  n labelled balls, m unlabelled urns	partitions of {1, ... , n} into ≤ m parts	n pigeons in m holes	partitions of {1, ... , n} into m parts
  n unlabelled balls, m unlabelled urns	partitions of n into ≤ m parts	n pigeons in m holes	partitions of n into m parts

• "balls and urns" is sometimes modeled as "stars and bars" [2]

Knuth's "The Art of Computer Programming"

• 7.2. Generating All Possibilities
• 7.2.1. Generating Basic Combinatorial Patterns
• 7.2.1.1. Generating all n-tuples
• 7.2.1.2. Generating all permutations
• 7.2.1.3. Generating all combinations
• 7.2.1.4. Generating all partitions
• 7.2.1.5. Generating all set partitions
• 7.2.1.6. Generating all trees
• 7.2.1.7. History and further references